{"appState":{"pageLoadApiCallsStatus":true},"categoryState":{"relatedCategories":{"headers":{"timestamp":"2022-05-27T12:31:26+00:00"},"categoryId":33721,"data":{"title":"Algebra","slug":"algebra","image":{"src":null,"width":0,"height":0},"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"parentCategory":{"categoryId":33720,"title":"Math","slug":"math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"}},"childCategories":[],"description":"Master algebra and algebra II with quick breakdowns of key math skills, plus practice problem sets and other cool stuff.","relatedArticles":{"self":"https://dummies-api.dummies.com/v2/articles?category=33721&offset=0&size=5"}},"_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"relatedCategoriesLoadedStatus":"success"},"listState":{"list":{"count":10,"total":206,"items":[{"headers":{"creationTime":"2016-03-27T16:56:11+00:00","modifiedTime":"2022-04-28T15:29:39+00:00","timestamp":"2022-04-28T18:01:07+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Algebra Workbook For Dummies Cheat Sheet","strippedTitle":"algebra workbook for dummies cheat sheet","slug":"algebra-workbook-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"Use these formulas, patterns, and procedures to simplify expressions and solve equations for all your algebra needs.","noIndex":0,"noFollow":0},"content":"Formulas, patterns, and procedures used for simplifying expressions and solving equations are basic to algebra. Use the equations, shortcuts, and formulas you find for quick reference. This Cheat Sheet offers basic information and short explanations (and some words of advice on traps to avoid).","description":"Formulas, patterns, and procedures used for simplifying expressions and solving equations are basic to algebra. Use the equations, shortcuts, and formulas you find for quick reference. This Cheat Sheet offers basic information and short explanations (and some words of advice on traps to avoid).","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. 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By applying the order of operations, you maintain that value.</p>\n<p>Apply the order of operations when no grouping symbols, such as parentheses, interrupt. When more of one level occurs in a problem, do them in order from left to right. When you perform operations on algebraic expressions and you have a choice between one or more operations to perform, use the following order:</p>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">Powers and roots</p>\n</li>\n<li>\n<p class=\"first-para\">Multiplication and division</p>\n</li>\n<li>\n<p class=\"first-para\">Addition and subtraction</p>\n</li>\n</ol>\n<p>These rules are interrupted if the problem has grouping symbols. You first need to perform operations in grouping symbols, such as ( ), { }, [ ] , above and below fraction lines, and inside radicals.</p>\n"},{"title":"Rules of exponents","thumb":null,"image":null,"content":"<p>Exponents are shorthand for repeated multiplication. The rules for performing operations involving exponents allow you to change multiplication and division expressions with the same base to something simpler. Remember that in xa the x is the base and the “a�? is the exponent.</p>\n<p>Assume that neither x nor y are equal to zero:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/200147.image0.jpg\" alt=\"image0.jpg\" width=\"171\" height=\"400\" /></p>\n"},{"title":"Selected math formulas step by step","thumb":null,"image":null,"content":"<p>Algebraic formulas make life (and algebra) simpler. You save time by not having to perform more complicated tasks. When using the formulas, use the appropriate rules for simplifying algebraic expressions. Also watch out for pitfalls; to help you, an asterisk (*) appears beside steps where errors are easy to make.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/200149.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"255\" /><br />\n<img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/200150.image1.jpg\" alt=\"image1.jpg\" width=\"535\" height=\"655\" /></p>\n"},{"title":"Factoring special problems","thumb":null,"image":null,"content":"<p>Binomials, their powers, and their products with selected trinomials occur frequently in algebraic processes. By using the patterns shown here, you save time and reduce the opportunity for errors.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/200143.image0.jpg\" alt=\"image0.jpg\" width=\"400\" height=\"274\" /></p>\n"},{"title":"Formulas for common geometric shapes","thumb":null,"image":null,"content":"<p>Depending on the algebra problem, you&#8217;ll need to know some geometry. The following represents some of the most common shapes in geometry and their formulas for perimeter, area, volume, surface areas, and circumference:</p>\n<table border=\"0\">\n<caption>Two-Dimensional Shapes</caption>\n<tbody>\n<tr>\n<th>Shape</th>\n<th>Perimeter/Circumference</th>\n<th>Area</th>\n</tr>\n<tr>\n<td>Rectangle</td>\n<td>P = 2(l + w)</td>\n<td>A = lw</td>\n</tr>\n<tr>\n<td>Square</td>\n<td>P = 4s</td>\n<td>A = s<sup>2</sup></td>\n</tr>\n<tr>\n<td>Triangle</td>\n<td>P = a + b + c</td>\n<td>A = 1/2bh</td>\n</tr>\n<tr>\n<td>Trapezoid</td>\n<td>P = a + b<sub>1</sub> + c + b<sub>2</sub></td>\n<td>A = 1/2h(b<sub>1</sub> + b<sub>2</sub>)</td>\n</tr>\n<tr>\n<td>Isosceles Trapezoid</td>\n<td>P = 2w + b<sub>1</sub> + b<sub>2</sub></td>\n<td>A = 1/2h(b<sub>1</sub> + b<sub>2</sub>)</td>\n</tr>\n<tr>\n<td>Circle</td>\n<td>C = πd = 2 π r</td>\n<td>A = π r<sup>2</sup></td>\n</tr>\n</tbody>\n</table>\n<table border=\"0\">\n<caption>Three-Dimensional Shapes</caption>\n<tbody>\n<tr>\n<th>Shape</th>\n<th>Surface Area</th>\n<th>Volume</th>\n</tr>\n<tr>\n<td>Box</td>\n<td>SA = 2lw + 2lh + 2wh</td>\n<td>V = lwh</td>\n</tr>\n<tr>\n<td>Sphere</td>\n<td>SA = 4 π r<sup>2</sup></td>\n<td>V = 4/3 π r<sup>3</sup></td>\n</tr>\n<tr>\n<td>Cylinder</td>\n<td>SA = 2 π r(r + h)</td>\n<td>V = π r<sup>2</sup>h</td>\n</tr>\n</tbody>\n</table>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-04-28T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":209000},{"headers":{"creationTime":"2016-03-27T16:49:18+00:00","modifiedTime":"2022-04-19T20:51:04+00:00","timestamp":"2022-04-20T00:01:09+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Algebra II: 1001 Practice Problems For Dummies Cheat Sheet","strippedTitle":"algebra ii: 1001 practice problems for dummies cheat sheet","slug":"1001-algebra-ii-practice-problems-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"This handy Cheat Sheet is a great reference for algebra problems, including equations, systems, graph lines, functions, and more.","noIndex":0,"noFollow":0},"content":"The best way to figure out how the different algebraic rules work and interact with one another is to practice with lots of problems. And Algebra II requires lots of practice. So be prepared to solve equations and systems, graph lines, tackle functions, and so much more.","description":"The best way to figure out how the different algebraic rules work and interact with one another is to practice with lots of problems. And Algebra II requires lots of practice. So be prepared to solve equations and systems, graph lines, tackle functions, and so much more.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":165288,"title":"Working with Radical and Rational Equations in Algebra II","slug":"working-with-radical-and-rational-equations-in-algebra-ii","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/165288"}},{"articleId":165267,"title":"Polynomial Functions and Equations in Algebra II","slug":"polynomial-functions-and-equations-in-algebra-ii","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/165267"}},{"articleId":165264,"title":"Systems of Linear Equations in Algebra II","slug":"systems-of-linear-equations-in-algebra-ii","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/165264"}}],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281912,"slug":"algebra-ii-1001-practice-problems-for-dummies-free-online-practice","isbn":"9781119883562","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"https://www.amazon.com/gp/product/1119883563/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119883563/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119883563-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119883563/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119883563/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/9781119883562-204x255.jpg","width":204,"height":255},"title":"Algebra II: 1001 Practice Problems For Dummies (+ Free Online Practice)","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"\n <p><b data-author-id=\"8985\">Mary Jane Sterling</b> taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.</p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119883562&quot;]}]\" id=\"du-slot-625f4d45eaf01\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119883562&quot;]}]\" id=\"du-slot-625f4d45eb88a\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":165288,"title":"Working with Radical and Rational Equations in Algebra II","slug":"working-with-radical-and-rational-equations-in-algebra-ii","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/165288"}},{"articleId":165267,"title":"Polynomial Functions and Equations in Algebra II","slug":"polynomial-functions-and-equations-in-algebra-ii","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/165267"}},{"articleId":165264,"title":"Systems of Linear Equations in Algebra II","slug":"systems-of-linear-equations-in-algebra-ii","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/165264"}}],"content":[{"title":"Working with radical and rational equations","thumb":null,"image":null,"content":"<p>A <i>radical</i> equation is one that starts out with a square root, cube root, or some other root and gets changed into another form to make the solving process easier. A <i>rational </i>equation is one that involves a fractional expression — usually with a polynomial in the numerator and denominator. Avoid these mistakes when working with radical or rational equations:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Forgetting to check for extraneous solutions</p>\n</li>\n<li>\n<p class=\"first-para\">Squaring a binomial incorrectly when squaring both sides to get rid of the radical</p>\n</li>\n<li>\n<p class=\"first-para\">Distributing correctly when writing equivalent fractions using a common denominator</p>\n</li>\n<li>\n<p class=\"first-para\">Eliminating solutions that create a 0 in the denominator</p>\n</li>\n</ul>\n"},{"title":"Polynomial functions and equations","thumb":null,"image":null,"content":"<p>In Algebra II, a polynomial function is one in which the coefficients are all real numbers, and the exponents on the variables are all whole numbers. A polynomial whose greatest power is 2 is called a <i>quadratic polynomial</i><i>;</i> if the highest power is 3, then it’s called a <i>cubic polynomial</i><i>.</i> A highest power of 4 earns the name <i>quartic</i> (not to be confused with quadratic), and a highest power of 5 is called <i>quintic</i><i>.</i></p>\n<p>When solving polynomial functions and equations, don’t let these common mistakes trip you up:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Forgetting to change the signs in the factored form when identifying x-intercepts</p>\n</li>\n<li>\n<p class=\"first-para\">Making errors when simplifying the terms in f(–x) applying Descartes’ rule of sign</p>\n</li>\n<li>\n<p class=\"first-para\">Not changing the sign of the divisor when using synthetic division</p>\n</li>\n<li>\n<p class=\"first-para\">Not distinguishing between curves that cross from those that just touch the x-axis at an intercept</p>\n</li>\n<li>\n<p class=\"first-para\">Graphing the incorrect end-behavior on the right and left of the graphs</p>\n</li>\n</ul>\n"},{"title":"Systems of linear equations","thumb":null,"image":null,"content":"<p>In Algebra II, a <i>linear equation</i> consists of variable terms whose exponents are always the number 1. When you have two variables, the equation can be represented by a line. With three terms, you can draw a plane to describe the equation. More than three variables is indescribable, because there are only three dimensions. When you have a system of linear equations, you can find the values of the variables that work for all the equations in the system — the common solutions. Sometimes there’s just one solution, sometimes many, and sometimes there’s no solution at all.</p>\n<p>When solving systems of linear equations, watch out for these mistakes:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Forgetting to change the signs in the factored form when identifying x-intercepts</p>\n</li>\n<li>\n<p class=\"first-para\">Making errors when simplifying the terms in f(–x) applying Descartes’ rule of sign</p>\n</li>\n<li>\n<p class=\"first-para\">Not changing the sign of the divisor when using synthetic division</p>\n</li>\n<li>\n<p class=\"first-para\">Not distinguishing between curves that cross from those that just touch the x-axis at an intercept</p>\n</li>\n<li>\n<p class=\"first-para\">Graphing the incorrect end-behavior on the right and left of the graphs</p>\n</li>\n</ul>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-01-28T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":207944},{"headers":{"creationTime":"2016-03-27T16:48:03+00:00","modifiedTime":"2022-04-12T18:48:58+00:00","timestamp":"2022-04-13T00:01:04+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Algebra II Workbook For Dummies Cheat Sheet","strippedTitle":"algebra ii workbook for dummies cheat sheet","slug":"algebra-ii-workbook-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"Learning some algebraic rules for various exponents, formulas, and equations will help you successfully solve problems in Algebra II.","noIndex":0,"noFollow":0},"content":"Learning some algebraic rules for various exponents, radicals, laws, binomials, formulas, and equations will help you successfully study and solve problems in an Algebra II course. You should also be able to recognize formulas to find slope, slope-intercept, distance, and midpoint (which are formulas from geometry) to help you through Algebra II.","description":"Learning some algebraic rules for various exponents, radicals, laws, binomials, formulas, and equations will help you successfully study and solve problems in an Algebra II course. You should also be able to recognize formulas to find slope, slope-intercept, distance, and midpoint (which are formulas from geometry) to help you through Algebra II.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":149411,"title":"Solving Equations with Complex Solutions","slug":"solving-equations-with-complex-solutions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/149411"}},{"articleId":149344,"title":"How to Put Equations of Parabolas in Standard Form","slug":"how-to-put-equations-of-parabolas-in-standard-form","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/149344"}}],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281941,"slug":"algebra-ii-workbook-for-dummies-3rd-edition","isbn":"9781119543114","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"https://www.amazon.com/gp/product/1119543118/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119543118/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119543118-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119543118/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119543118/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/algebra-ii-workbook-for-dummies-3rd-edition-cover-9781119543114-204x255.jpg","width":204,"height":255},"title":"Algebra II Workbook For Dummies, 3rd Edition","testBankPinActivationLink":"https://testbanks.wiley.com","bookOutOfPrint":false,"authorsInfo":"\n <p><b data-author-id=\"8985\">Mary Jane Sterling</b> taught junior high and high school math before embarking on her career as an instructor at Bradley University, where she taught for more than 35 years. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119543114&quot;]}]\" id=\"du-slot-625612c0d51b9\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119543114&quot;]}]\" id=\"du-slot-625612c0d5714\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":153121,"title":"Algebra: Rules of Exponents","slug":"algebra-rules-of-exponents","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/153121"}},{"articleId":153120,"title":"Linear Equations: How to Find Slope, <i>y</i>-Intercept, Distance, Midpoint","slug":"linear-equations-how-to-find-slope-y-intercept-distance-midpoint","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/153120"}},{"articleId":153075,"title":"Rewrite Absolute Value Equations as Linear Equations","slug":"rewrite-absolute-value-equations-as-linear-equations","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/153075"}},{"articleId":153020,"title":"9 Number Systems in Algebra to Know","slug":"9-number-systems-in-algebra-to-know","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/153020"}},{"articleId":153123,"title":"Algebra II: What Is the Binomial Theorem?","slug":"algebra-ii-what-is-the-binomial-theorem","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/153123"}},{"articleId":153137,"title":"Use the Properties of Proportions to Simplify Fractions","slug":"use-the-properties-of-proportions-to-simplify-fractions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/153137"}},{"articleId":153138,"title":"Algebra II: Raise Binomials to a Power","slug":"algebra-ii-raise-binomials-to-a-power","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/153138"}}],"content":[{"title":"Algebra: Rules of exponents","thumb":null,"image":null,"content":"<p>Exponents are shorthand for repeated multiplication. The rules for performing operations involving exponents allow you to change multiplication and division expressions with the same base into something simpler to work with. Remember that in <i>x</i><i><sup>a</sup></i>, the <i>x</i> is the <i>base</i> and the <i>a</i> is the <i>exponent</i>.</p>\n<p>Assume <i>x</i> ≠ 0:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425924.image0.png\" alt=\"image0.png\" width=\"73\" height=\"202\" /></p>\n"},{"title":"Linear equations: how to find slope, y-intercept, distance, midpoint","thumb":null,"image":null,"content":"<p>In algebra, linear equations means you&#8217;re dealing with straight lines. When you&#8217;re working with the <i>xy</i>-coordinate system, you can use the following formulas to find the slope, <i>y</i>-intercept, distance, and midpoint between two points.</p>\n<p>Consider the two points (<i>x</i><sub>1</sub>, <i>y</i><sub>1</sub>) and (<i>x</i><sub>2</sub>, <i>y</i><sub>2</sub>):</p>\n<blockquote><p><b>Slope of the line through the points:</b></p></blockquote>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425926.image0.png\" alt=\"image0.png\" width=\"72\" height=\"30\" /></p>\n<blockquote><p><b>Slope-intercept form of the line with <i>y</i>-intercept <i>b</i>:</b></p></blockquote>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425927.image1.png\" alt=\"image1.png\" width=\"68\" height=\"15\" /></p>\n<blockquote><p><b>Point-slope form of the line with slope <i>m</i>:</b></p></blockquote>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425928.image2.png\" alt=\"image2.png\" width=\"110\" height=\"15\" /></p>\n<blockquote><p><b>Distance formula:</b></p></blockquote>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425929.image3.png\" alt=\"image3.png\" width=\"168\" height=\"28\" /></p>\n<blockquote><p><b>Midpoint formula:</b></p></blockquote>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425930.image4.png\" alt=\"image4.png\" width=\"140\" height=\"27\" /></p>\n"},{"title":"Rewrite absolute value equations as linear equations","thumb":null,"image":null,"content":"<p>To work with an absolute value equation in algebra, you first need to rewrite it as a linear equation. The same goes for an absolute value inequality, which you rewrite as a linear inequality.</p>\n<p>When rewriting absolute value equations or inequalities, you drop the absolute value bars.</p>\n<blockquote><p>|<i>ax</i> + <i>b</i>| = <i>c</i><i> </i><i>→</i> <i>ax</i> + <i>b</i> = <i>c</i> or <i>ax</i> + <i>b</i> = –<i>c</i></p></blockquote>\n<blockquote><p>|<i>ax</i> + <i>b</i>| &gt; <i>c →</i> <i>ax</i> + <i>b</i> &gt; <i>c</i> or <i>ax</i> + <i>b</i> &gt; –<i>c</i></p></blockquote>\n<blockquote><p>|<i>ax</i> + <i>b</i>| &lt; <i>c →</i> –<i>c</i> &lt; <i>ax</i> + <i>b</i> &lt; <i>c</i></p></blockquote>\n"},{"title":"9 number systems in algebra to know","thumb":null,"image":null,"content":"<p>A <i>number system</i> in algebra is a set of numbers — and different number systems are used to solve different types of algebra problems. Number systems include real numbers, natural numbers, whole numbers, integers, rational numbers, irrational numbers, even numbers, and odd numbers.</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><b>Real numbers: </b>Real numbers comprise the full spectrum of numbers. They cover the gamut and can take on any form — fractions or whole numbers, decimal points or no decimal points. The full range of real numbers includes decimals that can go on forever. Real numbers are different from imaginary or complex numbers.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Natural numbers:</b> A natural number is a number that comes naturally. What numbers did you first use? Remember someone asking, &#8220;How old are you?&#8221; You proudly held up four fingers and said, &#8220;Four!&#8221; Natural numbers are greater than zero but don&#8217;t include fractions: 1, 2, 3, 4, 5, 6, 7, and so on, into infinity. You use natural numbers to count items and to make lists.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Whole numbers: </b>Whole numbers are just all the natural numbers plus a zero: 0, 1, 2, 3, 4, 5, and so on, into infinity. They act like natural numbers and are used when whole amounts (no fractions) are required. Zero can also indicate none.</p>\n<p class=\"child-para Remember\">Algebraic problems often require you to round the answer to the nearest whole number. This makes perfect sense when the problem involves people, cars, animals, houses, or anything that shouldn&#8217;t be cut into pieces.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Integers:</b> Integers incorporate all the whole numbers and their opposites (or additive inverses of the whole numbers). Integers can be described as being positive and negative whole numbers and 0: . . . –3, –2, –1, 0, 1, 2, 3, . . . .</p>\n<p class=\"child-para\">Integers are popular in algebra. When you solve a long, complicated problem and come up with an integer, you can be joyous because your answer is probably right. After all, it&#8217;s not a fraction! This doesn&#8217;t mean that answers in algebra can&#8217;t be fractions or decimals. It&#8217;s just that most textbooks and reference books try to stick with nice answers to increase the comfort level and avoid confusion.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Rational numbers: </b>Rational numbers are numbers that act rationally! In this case, acting rationally means that the decimal equivalent of the rational number behaves. The decimal ends somewhere, or it has a repeating pattern to it. That&#8217;s what constitutes &#8220;behaving.&#8221; Some examples of rational numbers with decimals that terminate include 2, 3.4, 5.77623, and –4.5.</p>\n<p class=\"child-para\">Some examples of rational numbers with decimals that repeat the same pattern include the following:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425933.image0.png\" alt=\"image0.png\" width=\"88\" height=\"15\" /><br />\n<img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425934.image1.png\" alt=\"image1.png\" width=\"88\" height=\"15\" /></p>\n<p class=\"child-para\">(The horizontal bar over the 164 and the 6 lets you know that these numbers repeat forever.) In all cases, rational numbers can be written as a fraction. They all have a fraction that they are equal to.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Irrational numbers:</b> Irrational numbers are real numbers that are not rational numbers. An irrational number cannot be written as a fraction, and decimal values for irrationals never end and never have a nice pattern to them. For example, pi, with its never-ending decimal places, is irrational.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Imaginary/complex numbers: </b>A number that isn&#8217;t real can be imaginary or complex. An imaginary number contains some multiple of <i>i</i>, which is the following:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425935.image2.png\" alt=\"image2.png\" width=\"34\" height=\"18\" /></p>\n<p class=\"child-para\">For example, 2 + 3<i>i</i> is a complex number.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Even numbers:</b> An even number is one that divides evenly by 2, such as 2, 4, 18, and 352.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Odd numbers:</b> An odd number is one that does not divide evenly by 2, such as 1, 3, 27, and 485.</p>\n</li>\n</ul>\n"},{"title":"Algebra II: What is the binomial theorem?","thumb":null,"image":null,"content":"<p>A <i>binomial</i> is a mathematical expression that has two terms. In algebra, people frequently raise binomials to powers to complete computations. The binomial theorem says that if <i>a</i> and <i>b</i> are real numbers and <i>n</i> is a positive integer, then</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425937.image0.png\" alt=\"image0.png\" width=\"507\" height=\"83\" /></p>\n<p>You can see the rule here, in the second line, in terms of the coefficients that are created using combinations. The powers on <i>a</i> start with <i>n</i> and decrease until the power is zero in the last term. That&#8217;s why you don&#8217;t see an <i>a</i> in the last term — it&#8217;s <i>a</i><sup>0</sup>, which is really a 1. The powers on <i>b</i> increase from <i>b</i><sup>0 </sup>until the last term, where it&#8217;s <i>b</i><i><sup>n</sup></i>. Notice that the power of <i>b</i> matches <i>k</i> in the combination.</p>\n"},{"title":"Use the properties of proportions to simplify fractions","thumb":null,"image":null,"content":"<p>In algebra, the properties of proportions come in handy when solving equations involving fractions. When you can, change an algebraic equation with fractions in it to a proportion for easy solving.</p>\n<p>If</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425939.image0.png\" alt=\"image0.png\" width=\"42\" height=\"24\" /></p>\n<p>then the following are all true:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425940.image1.png\" alt=\"image1.png\" width=\"42\" height=\"28\" /></p>\n<p>&nbsp;</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425941.image2.png\" alt=\"image2.png\" width=\"85\" height=\"28\" /></p>\n<p>&nbsp;</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425942.image3.png\" alt=\"image3.png\" width=\"85\" height=\"28\" /></p>\n<p>&nbsp;</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425943.image4.png\" alt=\"image4.png\" width=\"85\" height=\"36\" /></p>\n<p>A proportion is an equation involving two ratios (fractions) set equal to each other. The following equation is a proportion:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425944.image5.png\" alt=\"image5.png\" width=\"57\" height=\"27\" /></p>\n<p>Both fractions in that proportion reduce to</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425945.image6.png\" alt=\"image6.png\" width=\"13\" height=\"27\" /></p>\n<p>so it&#8217;s fairly easy to see how this statement is true.</p>\n<p>Proportions have some interesting, helpful, and easy-to-use properties. For example, in the following proportion,</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425946.image7.png\" alt=\"image7.png\" width=\"42\" height=\"24\" /></p>\n<p>the cross-products are equal: <em>a</em> <em>∙</em> <em>d = b</em> <em>∙</em> <em>c</em>.</p>\n<p>The reciprocals are equal (you can flip the fractions):</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425947.image8.png\" alt=\"image8.png\" width=\"42\" height=\"28\" /></p>\n<p>You can reduce the fractions vertically or horizontally: You can divide out factors that are common to both numerators or both denominators or the left fraction or the right fraction. (You can&#8217;t, however, divide out a factor from the numerator of one fraction and the denominator of the other.)</p>\n"},{"title":"Algebra II: Raise binomials to a power","thumb":null,"image":null,"content":"<p>A <i>binomial</i> is a mathematical expression that has two terms. In algebra, people frequently raise binomials to powers in order to solve equations. Here are some examples:</p>\n<blockquote><p>(<i>a</i> + <i>b</i>)<sup>0</sup> = 1</p></blockquote>\n<blockquote><p>(<i>a</i> + <i>b</i>)<sup>1</sup> = <i>a</i> + <i>b</i></p></blockquote>\n<blockquote><p>(<i>a</i> + <i>b</i>)<sup>2</sup> = <i>a</i><sup>2</sup> + 2<i>ab</i> + <i>b</i><sup>2</sup></p></blockquote>\n<blockquote><p>(<i>a</i> + <i>b</i>)<sup>3</sup> = <i>a</i><sup>3</sup> + 3<i>a</i><sup>2</sup><i>b</i> + 3<i>ab</i><sup>2</sup> + <i>b</i><sup>3</sup></p></blockquote>\n<blockquote><p>(<i>a</i> + <i>b</i>)<sup>4</sup> = <i>a</i><sup>4</sup> + 4<i>a</i><sup>3</sup><i>b</i> + 6<i>a</i><sup>2</sup><i>b</i><sup>2</sup> + 4<i>ab</i><sup>3</sup> + <i>b</i><sup>4</sup></p></blockquote>\n<blockquote><p>(<i>a</i> + <i>b</i>)<sup>5</sup> = <i>a</i><sup>5</sup> + 5<i>a</i><sup>4</sup><i>b</i> + 10<i>a</i><sup>3</sup><i>b</i><sup>2</sup> + 10<i>a</i><sup>2</sup><i>b</i><sup>3</sup> + 5<i>ab</i><sup>4</sup> + <i>b</i><sup>5</sup></p></blockquote>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-04-12T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":207689},{"headers":{"creationTime":"2016-03-27T16:49:35+00:00","modifiedTime":"2022-03-28T14:24:38+00:00","timestamp":"2022-03-28T18:01:06+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Algebra I: 1001 Practice Problems For Dummies Cheat Sheet","strippedTitle":"algebra i: 1001 practice problems for dummies cheat sheet","slug":"1001-algebra-i-practice-problems-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"As you're learning algebra, keep this handy guide nearby as a reference for factoring and solving basic equations.","noIndex":0,"noFollow":0},"content":"Algebra problems are easier to solve when you know the rules and formulas. Unlike other subjects where you can just read or listen and absorb the information sufficiently, math takes practice.\r\n\r\nThe only way to figure out how the different algebraic rules work and interact with one another is to get into the problems and get your hands dirty, so to speak. Be prepared to practice on lots of different problems.","description":"Algebra problems are easier to solve when you know the rules and formulas. Unlike other subjects where you can just read or listen and absorb the information sufficiently, math takes practice.\r\n\r\nThe only way to figure out how the different algebraic rules work and interact with one another is to get into the problems and get your hands dirty, so to speak. Be prepared to practice on lots of different problems.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":167671,"title":"Factoring in Algebra I","slug":"factoring-in-algebra-i","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/167671"}},{"articleId":167670,"title":"The Basics of Solving Algebra I Equations","slug":"the-basics-of-solving-algebra-i-equations","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/167670"}}],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281911,"slug":"algebra-i-1001-practice-problems-for-dummies-free-online-practice","isbn":"9781119883470","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"https://www.amazon.com/gp/product/1119883474/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119883474/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119883474-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119883474/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119883474/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/9781119883470-204x255.jpg","width":204,"height":255},"title":"Algebra I: 1001 Practice Problems For Dummies (+ Free Online Practice)","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"\n <p><b data-author-id=\"8985\">Mary Jane Sterling</b> taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.</p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119883470&quot;]}]\" id=\"du-slot-6241f7e23140a\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119883470&quot;]}]\" id=\"du-slot-6241f7e231ece\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":167671,"title":"Factoring in Algebra I","slug":"factoring-in-algebra-i","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/167671"}},{"articleId":167670,"title":"The Basics of Solving Algebra I Equations","slug":"the-basics-of-solving-algebra-i-equations","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/167670"}}],"content":[{"title":"Factoring in algebra","thumb":null,"image":null,"content":"<p>Factoring algebraic expressions is one of the most important techniques you need to practice. Not much else can be done in terms of solving equations, graphing functions and conics, and working on math applications if you can&#8217;t pull out a common factor and simplify an expression.</p>\n<p>Here are some tips for factoring in algebra:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Make sure you apply divisibility rules correctly.</p>\n</li>\n<li>\n<p class=\"first-para\">Write a prime factorization with the correct exponents on the prime factors.</p>\n</li>\n<li>\n<p class=\"first-para\">Check that the terms divided after dividing out a greatest common factor (GCF) don&#8217;t still have a common factor.</p>\n</li>\n<li>\n<p class=\"first-para\">Reduce only factors, not terms.</p>\n</li>\n<li>\n<p class=\"first-para\">Write fractional answers with correct grouping symbols to distinguish remaining factors.</p>\n</li>\n</ul>\n<h3>Factoring binomials</h3>\n<p>A binomial is an expression with two terms. The terms can be separated by addition or subtraction. You have four possibilities for factoring binomials:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Factor out a greatest common factor.</p>\n</li>\n<li>\n<p class=\"first-para\">Factor as the difference of perfect squares.</p>\n</li>\n<li>\n<p class=\"first-para\">Factor as the difference of perfect cubes.</p>\n</li>\n<li>\n<p class=\"first-para\">Factor as the sum of perfect cubes.</p>\n</li>\n</ul>\n<p>If one of these methods doesn&#8217;t work, then the binomial doesn&#8217;t factor by using real numbers.</p>\n<h3>Factoring quadratic trinomials</h3>\n<p>You can factor trinomials with the form <i>ax</i><sup>2</sup> + <i>bx</i> + <i>c</i> in one of two ways:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Factor out a greatest common factor.</p>\n</li>\n<li>\n<p class=\"first-para\">Find two binomials whose product is that trinomial.</p>\n</li>\n</ul>\n<p>When finding the two binomials whose product is a particular trinomial, you work from the factors of the constant term and the factors of the coefficient of the lead term to create a sum or difference that matches the coefficient of the middle term. This technique can be expanded to trinomials that have the same general format but with exponents that are multiples of the basic trinomial.</p>\n"},{"title":"The basics of solving algebra equations","thumb":null,"image":null,"content":"<p>One of the most common goals in algebra I is solve an equation. Solving an equation means to identify the number or numbers you can replace the variable with to make a true statement. You&#8217;ll find factoring and the multiplication property of zero to be your first approach, and then you&#8217;ll also have the quadratic formula to use on some of the more challenging second degree equations. Polynomials can be solved using synthetic division to help with the factoring.</p>\n<h3>Linear equations</h3>\n<p>Linear equations are of the form <i>ax</i> + <i>b</i> = <i>c</i><i>,</i> where <i>x</i> is some variable, and <i>a</i><i>,</i> <i>b</i><i>,</i> and <i>c</i> are real numbers. To solve a linear equation, you perform a series of <i>opposite</i><i>s</i><i>:</i></p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">If a number is added to the term containing <i>x</i><i>,</i> you subtract that number from both sides of the equation.</p>\n</li>\n<li>\n<p class=\"first-para\">If a number is subtracted from the term containing the variable, you add.</p>\n</li>\n<li>\n<p class=\"first-para\">If a number multiplies the variable, you divide.</p>\n</li>\n<li>\n<p class=\"first-para\">If a number divides the variable, you multiply.</p>\n</li>\n</ul>\n<p>Just be sure that whatever you do to one side of the equation, you also do to the other side. Think of the equation as two expressions pivoting on either side of a balance scale: You need to keep the sides at the same <i>weigh</i><i>t</i><i>.</i></p>\n<h3>Quadratic equations</h3>\n<p>A quadratic equation has the form <i>ax</i><sup>2</sup> + <i>bx</i> + <i>c</i> = 0. The equation can have two solutions, only one solution (a double root), or no solutions among the real numbers. Where no real solution occurs, imaginary numbers are brought into the picture. Quadratic equations are solved most easily when the trinomial factors, but the quadratic formula is also a nice means to finding solutions.</p>\n<h3>Polynomials</h3>\n<p>A polynomial is a smooth curve that goes on and on forever, from negative infinity to positive infinity. To solve a polynomial means to set the equation equal to 0 and determine which, if any, numbers create a true statement. Any numbers satisfying this equation give you important information: They tell you where the graph of the polynomial crosses or touches the <i>x</i><i>&#8211;</i>axis.</p>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-01-28T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":207994},{"headers":{"creationTime":"2016-03-27T16:54:08+00:00","modifiedTime":"2022-03-14T14:35:38+00:00","timestamp":"2022-03-14T18:01:09+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Linear Algebra For Dummies Cheat Sheet","strippedTitle":"linear algebra for dummies cheat sheet","slug":"linear-algebra-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"Must-know linear algebra concepts, including common numerical values of trig functions, basic algebraic properties and calculator commands.","noIndex":0,"noFollow":0},"content":"To study and solve linear algebra equations successfully, you need to know common numerical values of trig functions, what elements determine a vector space, basic algebraic properties, and general commands using graphing calculators to solve linear algebra problems.","description":"To study and solve linear algebra equations successfully, you need to know common numerical values of trig functions, what elements determine a vector space, basic algebraic properties, and general commands using graphing calculators to solve linear algebra problems.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":188400,"title":"Algebraic Properties You Should Know","slug":"algebraic-properties-you-should-know","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188400"}},{"articleId":188396,"title":"Commonly Used Values of Selected Trig Functions","slug":"commonly-used-values-of-selected-trig-functions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188396"}},{"articleId":188395,"title":"How to Meet Vector Space Requirements","slug":"how-to-meet-vector-space-requirements","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188395"}},{"articleId":188390,"title":"Calculator Commands for Linear Algebra","slug":"calculator-commands-for-linear-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188390"}}],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282354,"slug":"linear-algebra-for-dummies","isbn":"9780470430903","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"https://www.amazon.com/gp/product/0470430907/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/0470430907/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/0470430907-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/0470430907/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/0470430907/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/linear-algebra-for-dummies-cover-9780470430903-203x255.jpg","width":203,"height":255},"title":"Linear Algebra For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of numerous <i>For Dummies</i> books. She is a lecturer at Bradley University in Peoria, Illinois, where she has taught courses in algebra, calculus, and other mathematics topics for almost 30 years. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9780470430903&quot;]}]\" id=\"du-slot-622f82e543633\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9780470430903&quot;]}]\" id=\"du-slot-622f82e543d59\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":188396,"title":"Commonly Used Values of Selected Trig Functions","slug":"commonly-used-values-of-selected-trig-functions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188396"}},{"articleId":188395,"title":"How to Meet Vector Space Requirements","slug":"how-to-meet-vector-space-requirements","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188395"}},{"articleId":188400,"title":"Algebraic Properties You Should Know","slug":"algebraic-properties-you-should-know","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188400"}},{"articleId":188390,"title":"Calculator Commands for Linear Algebra","slug":"calculator-commands-for-linear-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188390"}}],"content":[{"title":"Commonly used values of selected trig functions","thumb":null,"image":null,"content":"<p>When performing transformations in trig functions, such as rotations, you need to use the numerical values of these functions. Here are some of the more commonly used angles.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/249344.image0.png\" alt=\"image0.png\" width=\"441\" height=\"347\" /></p>\n"},{"title":"How to meet vector space requirements","thumb":null,"image":null,"content":"<p>In linear algebra, a set of elements is termed a <em>vector space</em> when particular requirements are met. For example, let a set consist of vectors <strong>u</strong>, <strong>v</strong>, and <strong>w</strong>. Also let <em>k</em> and <em>l</em> be real numbers, and consider the defined operations of ⊕ and ⊗. The set is a vector space if, under the operation of ⊕, it meets the following requirements:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><strong>Closure. u</strong> ⊕ <strong>v</strong> is in the set.</p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Commutativity. u</strong> ⊕ <strong>v = v</strong> ⊕ <strong>u.</strong></p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Associativity. u</strong> ⊕ (<strong>v</strong> ⊕ <strong>w</strong>) = (<strong>u</strong> ⊕ <strong>v</strong>) ⊕ <strong>w.</strong></p>\n</li>\n<li>\n<p class=\"first-para\"><strong>An identity element 0. u</strong> ⊕ <strong>0</strong> = <strong>0</strong> ⊕ <strong>u</strong> = <strong>u</strong> for any element <strong>u.</strong></p>\n</li>\n<li>\n<p class=\"first-para\"><strong>An inverse element −u. u</strong> ⊕ <strong>−u</strong> = <strong>−u</strong> ⊕<strong> u</strong> = <strong>0</strong></p>\n</li>\n</ul>\n<p>Under the operation of ⊗, the set is a vector space if it meets the following requirements:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><strong>Closure.</strong> <em>k</em> ⊗ <strong>u</strong> is in the set.</p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Distribution over a vector sum.</strong> <em>k</em> ⊗ (<strong>u</strong> ⊕ <strong>v</strong>) = <em>k</em> ⊗ <strong>u</strong> ⊕ <em>k</em>⊗ <strong>v</strong>.</p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Distribution over a scalar sum.</strong> (<em>k</em> + <em>l</em>) ⊗ <strong>u</strong> = <em>k</em> ⊗<strong>u</strong> ⊕ <em>l</em> ⊗ <strong>u.</strong></p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Associativity of a scalar product.</strong> <em>k</em> ⊗ (<em>l</em> ⊗ <strong>u</strong>) = (<em>kl</em>) ⊗ <strong>u.</strong></p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Multiplication by the scalar identity.</strong> 1 ⊗ <strong>u</strong> = <strong>u.</strong></p>\n</li>\n</ul>\n"},{"title":"Algebraic properties you should know","thumb":null,"image":null,"content":"<p>You can use a number of properties when working with linear algebraic expressions, including the commutative, associative, and distributive properties of addition and multiplication, as well as identities and inverses in addition and multiplication:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/249347.image0.png\" alt=\"image0.png\" width=\"420\" height=\"347\" /></p>\n"},{"title":"Calculator commands for linear algebra","thumb":null,"image":null,"content":"<p>Graphing calculators are wonderful tools for helping you solve linear algebra processes; they allow you to drain battery power rather than brain power. Since there is a wide variety of graphing calculators out there, the following are general instructions for help with linear algebra that apply to most graphing calculators:</p>\n<h2>To solve systems of equations by graphing:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Write each equation in <i>y</i> = <i>mx</i> + <i>b</i> form.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Insert equations in the <i>y</i>-menu.</p>\n</li>\n<li>\n<p class=\"first-para\">3. Graph the lines.</p>\n</li>\n<li>\n<p class=\"first-para\">4. Use the Intersection tool to get the answer.</p>\n</li>\n</ol>\n<h2>To add or subtract matrices:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into the matrices A and B.</p>\n</li>\n<li>\n<p class=\"first-para\">2. With a new screen, press [A] + [B] or [A] – [B], and press Enter.</p>\n</li>\n</ol>\n<h2>To multiply by a scalar:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into the matrix A.</p>\n</li>\n<li>\n<p class=\"first-para\">2. With a new screen, press the scalar and multiply: <i>k</i> * [A], and press Enter.</p>\n</li>\n</ol>\n<h2>To multiply two matrices together:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into the matrices A and B.</p>\n</li>\n<li>\n<p class=\"first-para\">2. With a new screen, press [A] * [B], and press Enter.</p>\n</li>\n</ol>\n<h2>To switch rows:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into a matrix.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Use <i>row swap</i>: <i>rowSwap</i> ([matrix name], first row, second row), and press Enter.</p>\n</li>\n</ol>\n<h2>To add two rows together:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into a matrix.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Use <i>row addition</i>: &#8220;<i>row</i> +&#8221;, ([matrix name], row to be added to target row, target row), and press Enter.</p>\n</li>\n</ol>\n<h2>To add the multiple of one row to another:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into a matrix.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Use row <i>sum</i>&#8211;<i>of</i>&#8211;<i>multiple</i>: &#8220;*<i>row</i> +&#8221;, (multiplier, [matrix name], row being multiplied, target row having multiple added to it), and press Enter.</p>\n</li>\n</ol>\n<h2>To multiply a row by a scalar:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into a matrix.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Use <i>row</i> <i>multiple</i>: &#8220;*row&#8221; (multiplier, [matrix name], row), and press Enter.</p>\n</li>\n</ol>\n<h2>To create an echelon form:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into a matrix.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Use <i>row</i>&#8211;<i>echelon</i> <i>form</i>: <i>ref</i> ([matrix name]) or <i>reduced row-echelon form: rref</i> ([matrix name]), and press Enter.</p>\n</li>\n</ol>\n<h2>To raise a matrix to a power:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into a matrix.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Use the caret operation with power, <i>p</i>: [matrix name] ^ <i>p</i>, and press Enter.</p>\n</li>\n</ol>\n<h2>To find inverses:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into a matrix.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Use the reciprocal operation, <i>x</i><sup>−1</sup>: [matrix name]<sup>−1</sup>, and press Enter.</p>\n</li>\n</ol>\n<h2>To solve systems of linear equations:</h2>\n<p>(This only works when the system has a single solution; it fails when the matrix A is singular.)</p>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Write each equation with the variables in the same order and the constant on the other side of the equation sign.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Create a matrix A, whose elements are the coefficients of the variables.</p>\n</li>\n<li>\n<p class=\"first-para\">3. Create a matrix B, whose elements are the constants.</p>\n</li>\n<li>\n<p class=\"first-para\">4. Press, A<sup>−1</sup> * B, and press Enter.</p>\n</li>\n</ol>\n<p>The resulting vector has the values of the variables, in order.</p>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-03-14T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":208656},{"headers":{"creationTime":"2016-03-27T16:57:12+00:00","modifiedTime":"2022-02-24T20:15:42+00:00","timestamp":"2022-02-25T00:01:04+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Algebra I For Dummies Cheat Sheet","strippedTitle":"algebra i for dummies cheat sheet","slug":"algebra-i-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"Take the stress out of learning algebra by mastering the rules and formulas.","noIndex":0,"noFollow":0},"content":"Algebra problems are easier to solve when you know the rules and formulas. Memorizing key algebra formulas will speed up your work, too. And if you know the rules of divisibility and the order of operations, you'll be able to solve algebra problems without getting stressed.","description":"Algebra problems are easier to solve when you know the rules and formulas. Memorizing key algebra formulas will speed up your work, too. And if you know the rules of divisibility and the order of operations, you'll be able to solve algebra problems without getting stressed.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":193760,"title":"Algebra's Rules of Divisibility","slug":"algebras-rules-of-divisibility","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/193760"}},{"articleId":193749,"title":"Algebra Formulas Worth Memorizing","slug":"algebra-formulas-worth-memorizing","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/193749"}},{"articleId":193748,"title":"Order of Operations in Algebra","slug":"order-of-operations-in-algebra-2","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/193748"}}],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281938,"slug":"algebra-i-for-dummies-2nd-edition","isbn":"9781119293576","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"https://www.amazon.com/gp/product/111929357X/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/111929357X/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/111929357X-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/111929357X/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/111929357X/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/algebra-i-for-dummies-2nd-edition-cover-9781119293576-202x255.jpg","width":202,"height":255},"title":"Algebra I For Dummies, 2nd Edition","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"\n <p><b data-author-id=\"8985\">Mary Jane Sterling</b> taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.</p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293576&quot;]}]\" id=\"du-slot-62181c4092878\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293576&quot;]}]\" id=\"du-slot-62181c4093205\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":193749,"title":"Algebra Formulas Worth Memorizing","slug":"algebra-formulas-worth-memorizing","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/193749"}},{"articleId":193760,"title":"Algebra's Rules of Divisibility","slug":"algebras-rules-of-divisibility","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/193760"}},{"articleId":193748,"title":"Order of Operations in Algebra","slug":"order-of-operations-in-algebra-2","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/193748"}}],"content":[{"title":"Algebra formulas worth memorizing ","thumb":null,"image":null,"content":"<div id=\"article-content-wrapper\" data-v-89b5964a=\"\" data-v-76ccd6ad=\"\">\n<article class=\"article-content w-100\" data-v-89b5964a=\"\" data-testid=\"articleContent\">Some formulas occur frequently when you&#8217;re doing algebraic manipulations and working through mathematical applications. You&#8217;ll find ways to use these algebra formulas even when you&#8217;re doing something other than algebra, such as planning a garden or a road trip. Of course, you need to know what the letters and symbols in the formula mean, so both the formulas and the explanations are in this table.</p>\n<div class=\"img-div\"><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/175704.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"567\" /></div>\n</article>\n</div>\n<div class=\"about-article-section\" data-v-2c35db38=\"\" data-v-76ccd6ad=\"\">\n<h2 class=\"h3 section-title\" data-v-2c35db38=\"\" data-testid=\"aboutArticleHead\"></h2>\n</div>\n"},{"title":"Algebra's rules of divisibility ","thumb":null,"image":null,"content":"<p>In algebra, knowing the rules of divisibility can help you solve faster. When factoring algebraic expressions to solve equations, you need to be able to pull out the greatest factor. You also need common factors when reducing algebraic fractions. The rules of divisibility help you find the common factors and change the algebraic expressions so that they&#8217;re put in a more workable form.</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><b>Divisibility by 2:</b> A number is divisible by 2 if the last digit in the number is 0, 2, 4, 6, or 8.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 3:</b> A number is divisible by 3 if the sum of the digits in the number is divisible by 3.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 4:</b> A number is divisible by 4 if the last two digits in the number form a number divisible by 4.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 5:</b> A number is divisible by 5 if the last digit is 0 or 5.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 6:</b> A number is divisible by 6 if it is divisible by both 2 and 3.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 8:</b> A number is divisible by 8 if the last three digits form a number divisible by 8.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 9:</b> A number is divisible by 9 if the sum of the digits of the number is divisible by 9.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 10:</b> A number is divisible by 10 if it ends in 0.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 11:</b> A number is divisible by 11 if the sums of the alternate digits are different by 0, 11, 22, or 33, or any two-digit multiple of 11. In other words, say you have a six-digit number: Add up the first, third, and fifth digits — the odd ones. Then add the digits in the even places — second, fourth, and sixth. Then subtract the smaller of those totals from the larger total, and if the answer is a multiple of 11, the original number is divisible by 11.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 12: </b>A number is divisible by 12 if the last two digits form a number divisible by 4 and if the sum of the digits is divisible by 3.</p>\n</li>\n</ul>\n"},{"title":"Order of operations in algebra ","thumb":null,"image":null,"content":"<p>Solve algebra problems correctly by following the order of operations. When performing more than one operation on an algebraic expression, work out the operations and signs in the following order:</p>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">First calculate powers and roots.</p>\n</li>\n<li>\n<p class=\"first-para\">Then perform all multiplication and division.</p>\n</li>\n<li>\n<p class=\"first-para\">Finish with addition and subtraction.</p>\n</li>\n</ol>\n<p>If you have more than two operations of the same level, do them in order from left to right, following the order of operations.</p>\n<p>For example, to solve 24 ÷ 3 + 11 – 3<sup>2</sup> × 2, you would</p>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">First calculate powers and roots.</p>\n<p class=\"child-para\">This problem doesn&#8217;t have any roots, but it does have one power, 3<sup>2</sup>. You know that 3<sup>2</sup> = 3 × 3 = 9. Substitute 9 into the problem, and you get 24 ÷ 3 + 11 – 9 × 2.</p>\n</li>\n<li>\n<p class=\"first-para\">Then perform all multiplication and division.</p>\n<p class=\"child-para\">Working left to right, 24 ÷ 3 = 8 and 9 × 2 = 18. Substitute those numbers into the problem, and you have 8 + 11 – 18.</p>\n</li>\n<li>\n<p class=\"first-para\">Finish with addition and subtraction.</p>\n<p class=\"child-para\">So, you end up with 8 + 11 – 18 = 19 – 18 = 1.</p>\n</li>\n</ol>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Two years","lifeExpectancySetFrom":"2022-02-24T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":209219},{"headers":{"creationTime":"2016-03-27T16:47:10+00:00","modifiedTime":"2022-01-21T18:01:48+00:00","timestamp":"2022-02-24T17:07:25+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Algebra II For Dummies Cheat Sheet","strippedTitle":"algebra ii for dummies cheat sheet","slug":"algebra-ii-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"This handy guide includes the algebraic equations for multiplying binomials, dealing with radicals, finding the sum of sequences, and more.","noIndex":0,"noFollow":0},"content":"Algebra is all about formulas, equations, and graphs. You need algebraic equations for multiplying binomials, dealing with radicals, finding the sum of sequences, and graphing the intersections of cones and planes. You also get to deal with logarithms, you lucky Algebra II user!","description":"Algebra is all about formulas, equations, and graphs. You need algebraic equations for multiplying binomials, dealing with radicals, finding the sum of sequences, and graphing the intersections of cones and planes. You also get to deal with logarithms, you lucky Algebra II user!","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":196396,"title":"Algebraic Permutations and Combinations","slug":"algebraic-permutations-and-combinations","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196396"}},{"articleId":196395,"title":"Algebra’s Quadratic Formula","slug":"algebras-quadratic-formula","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196395"}},{"articleId":196394,"title":"Rules for Radicals — the Algebraic Kind","slug":"rules-for-radicals-the-algebraic-kind","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196394"}},{"articleId":196389,"title":"Algebra Equations for Multiplying Binomials","slug":"algebra-equations-for-multiplying-binomials","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196389"}},{"articleId":196383,"title":"Cramer’s Rule for Linear Algebra","slug":"cramers-rule-for-linear-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196383"}}],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281940,"slug":"algebra-ii-for-dummies-2nd-edition","isbn":"9781119543145","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"https://www.amazon.com/gp/product/1119543142/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119543142/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119543142-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119543142/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119543142/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/algebra-ii-for-dummies-2nd-edition-cover-9781119543145-203x255.jpg","width":203,"height":255},"title":"Algebra II For Dummies, 2nd Edition","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"\n <p><b data-author-id=\"8985\">Mary Jane Sterling</b> taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.</p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119543145&quot;]}]\" id=\"du-slot-6217bb4db480a\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119543145&quot;]}]\" id=\"du-slot-6217bb4db5182\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":196389,"title":"Algebra Equations for Multiplying Binomials","slug":"algebra-equations-for-multiplying-binomials","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196389"}},{"articleId":196395,"title":"Algebra’s Quadratic Formula","slug":"algebras-quadratic-formula","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196395"}},{"articleId":196394,"title":"Rules for Radicals — the Algebraic Kind","slug":"rules-for-radicals-the-algebraic-kind","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196394"}},{"articleId":196381,"title":"Algebra’s Laws of Logarithms","slug":"algebras-laws-of-logarithms","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196381"}},{"articleId":143332,"title":"Standard Equations of Algebraic Conics","slug":"standard-equations-of-algebraic-conics","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/143332"}},{"articleId":196383,"title":"Cramer’s Rule for Linear Algebra","slug":"cramers-rule-for-linear-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196383"}},{"articleId":188291,"title":"Using Algebra to Find the Sums of Sequences","slug":"using-algebra-to-find-the-sums-of-sequences","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188291"}},{"articleId":196396,"title":"Algebraic Permutations and Combinations","slug":"algebraic-permutations-and-combinations","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196396"}},{"articleId":196382,"title":"Eight Basic Algebraic Curves","slug":"eight-basic-algebraic-curves","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196382"}}],"content":[{"title":"Algebraic equations for multiplying binomials","thumb":null,"image":null,"content":"<p>In algebra, multiplying binomials is easier if you recognize their patterns. You multiply the sum and difference of binomials and multiply by squaring and cubing to find some of the special products in algebra. See if you can spot the patterns in these equations:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Sum and difference: (<em>a</em> + <em>b</em>)(<em>a</em> – <em>b</em>) = <em>a</em><sup>2</sup> – <em>b</em><sup>2</sup></p>\n</li>\n<li>\n<p class=\"first-para\">Binomial squared: (<em>a</em> + <em>b</em>)<sup>2</sup> = <em>a</em><sup>2</sup> + 2<em>ab</em> + <em>b</em><sup>2</sup></p>\n</li>\n<li>\n<p class=\"first-para\">Binomial cubed: (<em>a</em> + <em>b</em>)<sup>3</sup> = <em>a</em><sup>3</sup> + 3<em>a</em><sup>2</sup><em>b </em>+ 3<em>ab</em><sup>2</sup> + <em>b</em><sup>3</sup></p>\n</li>\n</ul>\n"},{"title":"Algebra's quadratic formula","thumb":null,"image":null,"content":"<p>You can find solutions for quadratic equations by factoring, completing the square, guessing, or everyone&#8217;s favorite — using the quadratic formula. The best thing about the quadratic formula (unlike the other methods) is that it always works.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/132935.image0.jpg\" alt=\"image0.jpg\" width=\"404\" height=\"134\" /></p>\n"},{"title":"Rules for algebraic radicals","thumb":null,"image":null,"content":"<p>Working with radicals can be troublesome, but these equivalences keep algebraic radicals from running amok. Finding the root of product or quotient or a fractional exponent is simple with these formulas; just be sure that the numbers replacing the factors <i>a</i> and <i>b</i> are positive.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/132937.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"213\" /></p>\n"},{"title":"Algebra’s laws of logarithms","thumb":null,"image":null,"content":"<p>Logarithms help you add instead of multiply. The algebra formulas here make it easy to find equivalence, the logarithm of a product, quotient, power, reciprocal, base, and the log of 1.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/132939.image0.jpg\" alt=\"image0.jpg\" width=\"260\" height=\"400\" /></p>\n"},{"title":"Standard equations of algebraic conics","thumb":null,"image":null,"content":"<p>Conics are curved algebraic forms that come from slicing a cone with a plane. Use these equations to graph algebraic conics, such as circles, ellipses, parabolas, and hyperbolas:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/477053.image0.jpg\" alt=\"image0.jpg\" width=\"499\" height=\"400\" /></p>\n"},{"title":"Cramer's rule for linear algebra","thumb":null,"image":null,"content":"<p>Named for Gabriel Cramer, Cramer’s Rule provides a solution for a system of two linear algebraic equations in terms of determinants — the numbers associated with a specific, square matrix.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/132926.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"49\" /></p>\n"},{"title":"Using algebra to find the sums of sequences","thumb":null,"image":null,"content":"<p>Algebra can help you add a series of numbers (the sum of sequences) more quickly than you would be able to with straight addition. Adding integers, squares, cubes, and terms in an arithmetic or geometric sequence is simple with these algebraic formulas:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/254965.image0.png\" alt=\"image0.png\" width=\"447\" height=\"436\" /></p>\n"},{"title":"Algebraic Permutations and Combinations","thumb":null,"image":null,"content":"<p>In algebra, you use permutations to count the number of subsets of a larger set. Use permutations when order is necessary. With combinations, you can count the number of subsets when order doesn&#8217;t matter. The formulas you need are</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/132930.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"209\" /></p>\n"},{"title":"Eight basic algebraic curves","thumb":null,"image":null,"content":"<p>Algebra is all about graphing relationships, and the curve is one of the most basic shapes used. Here&#8217;s a look at eight of the most frequently used graphs.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/132932.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"886\" /></p>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-01-21T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":207507},{"headers":{"creationTime":"2016-03-26T07:10:03+00:00","modifiedTime":"2021-12-21T20:24:01+00:00","timestamp":"2022-02-24T17:07:18+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Convert a Circle Equation to the Standard Form","strippedTitle":"convert a circle equation to the standard form","slug":"convert-a-circle-equation-to-the-standard-form","canonicalUrl":"","seo":{"metaDescription":"Learn how to sketch the graph of a circle by using its equation in the standard form. All you need is the circle's center and radius.","noIndex":0,"noFollow":0},"content":"When the equation of a circle appears in the standard form, it provides you with all you need to know about the circle: its center and radius. With these two bits of information, you can sketch the graph of the circle.\r\n\r\nThe equation <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + 6<i>x</i> – 4<i>y</i> – 3 = 0, for example, is the equation of a circle. You can change this equation to the standard form by completing the square for each of the variables. Just follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Change the order of the terms so that the <i>x</i>'s and <i>y</i>'s are grouped together and the constant appears on the other side of the equal sign.</p>\r\n<p class=\"child-para\">Leave a space after the groupings for the numbers that you need to add:</p>\r\n<p class=\"child-para\"><i></i><i>x</i><sup>2</sup> + 6<i>x</i> _____ + <i>y</i><sup>2</sup> – 4<i>y</i> _____ = 3</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Complete the square for each variable, adding the number that creates perfect square trinomials.</p>\r\n<p class=\"child-para\">In the case of the <i>x</i>'s, you add 9, and with the <i>y</i>'s, you add 4. Don't forget to also add 9 and 4 to the right:</p>\r\n<p class=\"child-para\"><i></i><i>x</i><sup>2</sup> + 6<i>x</i> + 9 + <i>y</i><sup>2</sup> – 4<i>y</i> + 4 = 3 + 9 + 4</p>\r\n<p class=\"child-para\">When it's simplified, you have <i>x</i><sup>2</sup> + 6<i>x</i> + 9 + <i>y</i><sup>2</sup> – 4<i>y</i> + 4 = 16</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Factor each perfect square trinomial.</p>\r\n<p class=\"child-para\">The standard form for the equation of this circle is (<i>x</i> + 3)<sup>2</sup> + (<i>y</i> – 2)<sup>2</sup> = 16.</p>\r\n</li>\r\n</ol>\r\nThe circle has its center at the point (–3, 2) and has a radius of 4 (the square root of 16). To sketch this circle, you locate the point (–3, 2) and then count 4 units up, down, left, and right; sketch in a circle that includes those points. The figure below shows you the way.\r\n<div class=\"imageBlock\" style=\"width: 350px;\">\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/501757.image0.png\" alt=\"With the center, radius, and a compass, you too can sketch this circle.\" width=\"350\" height=\"212\" />\r\n<div class=\"imageCaption\">With the center, radius, and a compass, you too can sketch this circle.</div>\r\n</div>","description":"When the equation of a circle appears in the standard form, it provides you with all you need to know about the circle: its center and radius. With these two bits of information, you can sketch the graph of the circle.\r\n\r\nThe equation <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + 6<i>x</i> – 4<i>y</i> – 3 = 0, for example, is the equation of a circle. You can change this equation to the standard form by completing the square for each of the variables. Just follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Change the order of the terms so that the <i>x</i>'s and <i>y</i>'s are grouped together and the constant appears on the other side of the equal sign.</p>\r\n<p class=\"child-para\">Leave a space after the groupings for the numbers that you need to add:</p>\r\n<p class=\"child-para\"><i></i><i>x</i><sup>2</sup> + 6<i>x</i> _____ + <i>y</i><sup>2</sup> – 4<i>y</i> _____ = 3</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Complete the square for each variable, adding the number that creates perfect square trinomials.</p>\r\n<p class=\"child-para\">In the case of the <i>x</i>'s, you add 9, and with the <i>y</i>'s, you add 4. Don't forget to also add 9 and 4 to the right:</p>\r\n<p class=\"child-para\"><i></i><i>x</i><sup>2</sup> + 6<i>x</i> + 9 + <i>y</i><sup>2</sup> – 4<i>y</i> + 4 = 3 + 9 + 4</p>\r\n<p class=\"child-para\">When it's simplified, you have <i>x</i><sup>2</sup> + 6<i>x</i> + 9 + <i>y</i><sup>2</sup> – 4<i>y</i> + 4 = 16</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Factor each perfect square trinomial.</p>\r\n<p class=\"child-para\">The standard form for the equation of this circle is (<i>x</i> + 3)<sup>2</sup> + (<i>y</i> – 2)<sup>2</sup> = 16.</p>\r\n</li>\r\n</ol>\r\nThe circle has its center at the point (–3, 2) and has a radius of 4 (the square root of 16). To sketch this circle, you locate the point (–3, 2) and then count 4 units up, down, left, and right; sketch in a circle that includes those points. The figure below shows you the way.\r\n<div class=\"imageBlock\" style=\"width: 350px;\">\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/501757.image0.png\" alt=\"With the center, radius, and a compass, you too can sketch this circle.\" width=\"350\" height=\"212\" />\r\n<div class=\"imageCaption\">With the center, radius, and a compass, you too can sketch this circle.</div>\r\n</div>","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":207507,"title":"Algebra II For Dummies Cheat Sheet","slug":"algebra-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/207507"}},{"articleId":196396,"title":"Algebraic Permutations and Combinations","slug":"algebraic-permutations-and-combinations","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196396"}},{"articleId":196395,"title":"Algebra’s Quadratic Formula","slug":"algebras-quadratic-formula","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196395"}},{"articleId":196394,"title":"Rules for Radicals — the Algebraic Kind","slug":"rules-for-radicals-the-algebraic-kind","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196394"}},{"articleId":196389,"title":"Algebra Equations for Multiplying Binomials","slug":"algebra-equations-for-multiplying-binomials","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/196389"}}],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281940,"slug":"algebra-ii-for-dummies-2nd-edition","isbn":"9781119543145","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"https://www.amazon.com/gp/product/1119543142/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119543142/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119543142-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119543142/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119543142/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/algebra-ii-for-dummies-2nd-edition-cover-9781119543145-203x255.jpg","width":203,"height":255},"title":"Algebra II For Dummies, 2nd Edition","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"\n <p><b data-author-id=\"8985\">Mary Jane Sterling</b> taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.</p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":"Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. ","_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119543145&quot;]}]\" id=\"du-slot-6217bb46ecacc\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119543145&quot;]}]\" id=\"du-slot-6217bb46ed485\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2021-07-12T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":138679},{"headers":{"creationTime":"2016-03-26T21:47:54+00:00","modifiedTime":"2021-07-14T18:25:30+00:00","timestamp":"2022-02-24T17:06:47+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"How to Add and Subtract with Powers","strippedTitle":"how to add and subtract with powers","slug":"how-to-add-and-subtract-with-powers","canonicalUrl":"","seo":{"metaDescription":"","noIndex":0,"noFollow":0},"content":"To add or subtract with powers, both the variables and the <a href=\"https://www.dummies.com/education/math/algebra/using-exponents-to-simplify-equations/\" target=\"_blank\" rel=\"noopener\">exponents </a>of the variables must be the same. You perform the required operations on the <a href=\"https://www.dummies.com/education/math/algebra/how-to-work-with-coefficients/\" target=\"_blank\" rel=\"noopener\">coefficients</a>, leaving the variable and exponent as they are.\r\n\r\nWhen adding or subtracting with powers, the terms that combine always have exactly the same variables with exactly the same powers. These rules are true for multiplying and <a href=\"https://www.dummies.com/education/math/algebra/how-to-divide-exponents\" target=\"_blank\" rel=\"noopener\">dividing exponents</a> as well.\r\n\r\n<b><i>Example</i></b><b><i> 1</i></b><b><i>:</i></b><i> </i>\r\n\r\n<i>x</i> + <i>x</i> + <i>x</i> = 3<i>x</i>\r\n\r\nBecause the variables are the same (<i>x</i>) and the powers are the same (there are no exponents, so the exponents must be 1), you can add the variables.\r\n\r\n<b><i>Example</i></b><b><i> 2</i></b><b><i>:</i></b>\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165134.image0.png\" alt=\"image0.png\" width=\"254\" height=\"32\" />\r\n\r\nBecause the variables are the same (<i>x</i>) and the powers are the same (2), you can perform the required operations on the variables.\r\n\r\n<b><i>Example</i></b><b><i> 3</i></b><b><i>:</i></b>\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165135.image1.png\" alt=\"image1.png\" width=\"220\" height=\"32\" />\r\n\r\nAlthough the variables are the same (<i>x</i>), the powers are not the same (1, 2, 3, and 4). You can't simplify these terms because only the variables are the same, and both the variables and the powers need to be the same.\r\n\r\n<b><i>Example</i></b><b><i> 4</i></b><b><i>:</i></b><i> </i>\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165136.image2.png\" alt=\"image2.png\" width=\"310\" height=\"218\" />\r\n\r\nSometimes not all of the variables and powers will be the same — you may encounter a problem that has several groups of variables and powers that are not the same. In this case, you only <a href=\"https://www.dummies.com/education/math/algebra/how-to-add-and-subtract-with-powers/\" target=\"_blank\" rel=\"noopener\">add or subtract terms</a> whose variables and powers are the same. (Notice that the exponents are listed in order from highest to lowest. This is a common practice to make answers easy to compare.)","description":"To add or subtract with powers, both the variables and the <a href=\"https://www.dummies.com/education/math/algebra/using-exponents-to-simplify-equations/\" target=\"_blank\" rel=\"noopener\">exponents </a>of the variables must be the same. You perform the required operations on the <a href=\"https://www.dummies.com/education/math/algebra/how-to-work-with-coefficients/\" target=\"_blank\" rel=\"noopener\">coefficients</a>, leaving the variable and exponent as they are.\r\n\r\nWhen adding or subtracting with powers, the terms that combine always have exactly the same variables with exactly the same powers. These rules are true for multiplying and <a href=\"https://www.dummies.com/education/math/algebra/how-to-divide-exponents\" target=\"_blank\" rel=\"noopener\">dividing exponents</a> as well.\r\n\r\n<b><i>Example</i></b><b><i> 1</i></b><b><i>:</i></b><i> </i>\r\n\r\n<i>x</i> + <i>x</i> + <i>x</i> = 3<i>x</i>\r\n\r\nBecause the variables are the same (<i>x</i>) and the powers are the same (there are no exponents, so the exponents must be 1), you can add the variables.\r\n\r\n<b><i>Example</i></b><b><i> 2</i></b><b><i>:</i></b>\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165134.image0.png\" alt=\"image0.png\" width=\"254\" height=\"32\" />\r\n\r\nBecause the variables are the same (<i>x</i>) and the powers are the same (2), you can perform the required operations on the variables.\r\n\r\n<b><i>Example</i></b><b><i> 3</i></b><b><i>:</i></b>\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165135.image1.png\" alt=\"image1.png\" width=\"220\" height=\"32\" />\r\n\r\nAlthough the variables are the same (<i>x</i>), the powers are not the same (1, 2, 3, and 4). You can't simplify these terms because only the variables are the same, and both the variables and the powers need to be the same.\r\n\r\n<b><i>Example</i></b><b><i> 4</i></b><b><i>:</i></b><i> </i>\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165136.image2.png\" alt=\"image2.png\" width=\"310\" height=\"218\" />\r\n\r\nSometimes not all of the variables and powers will be the same — you may encounter a problem that has several groups of variables and powers that are not the same. In this case, you only <a href=\"https://www.dummies.com/education/math/algebra/how-to-add-and-subtract-with-powers/\" target=\"_blank\" rel=\"noopener\">add or subtract terms</a> whose variables and powers are the same. (Notice that the exponents are listed in order from highest to lowest. This is a common practice to make answers easy to compare.)","blurb":"","authors":[],"primaryCategoryTaxonomy":{"categoryId":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":0,"slug":null,"isbn":null,"categoryList":null,"amazon":null,"image":null,"title":null,"testBankPinActivationLink":null,"bookOutOfPrint":false,"authorsInfo":null,"authors":null,"_links":null},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-6217bb271d627\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-6217bb271df9a\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2021-07-12T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":194362},{"headers":{"creationTime":"2016-03-26T21:19:17+00:00","modifiedTime":"2021-07-14T18:20:27+00:00","timestamp":"2022-02-24T17:06:47+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"How to Combine Square Roots and Convert Square Roots to Exponents","strippedTitle":"how to combine square roots and convert square roots to exponents","slug":"how-to-convert-square-roots-to-exponents","canonicalUrl":"","seo":{"metaDescription":"Raise your knowledge of square roots and exponents to the zillionth power with this quick how-to article from Dummies.","noIndex":0,"noFollow":0},"content":"<h2 id=\"tab1\" >Combining square roots</h2>\r\nSquare roots, which use the radical symbol, are nonbinary operations — operations which involve just one number — that ask you, “What number times itself gives you this number under the radical?” Finding square roots is a relatively common operation in algebra, but working with and combining the roots isn’t always so clear.\r\n\r\nExpressions with radicals can be multiplied or divided as long as the root power or value under the radical is the same. Expressions with radicals cannot be added or subtracted unless both the root power and the value under the radical are the same.\r\n<p class=\"Remember\">When you find square roots, the symbol for that operation is called a radical. The root power refers to the number outside and to the upper left of the radical. If there is no number, you assume that the root power is 2.</p>\r\nWhen it comes to combining radicals, there are just a couple of simple rules to remember:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Addition and subtraction</b> can be performed if the root power <i>and</i><b><i> </i></b>the value under the radical are the same.</p>\r\n<p class=\"child-para\">Examples:</p>\r\n<p class=\"child-para\">These radicals cannot be combined because the operation is addition, and the value under the radical is not the same:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165223.image0.png\" alt=\"image0.png\" width=\"88\" height=\"38\" />\r\n<p class=\"child-para\">These radicals can be combined because the root power and the numbers under the radical are the same:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165224.image1.png\" alt=\"image1.png\" width=\"176\" height=\"38\" />\r\n<p class=\"child-para\">These radicals cannot be combined because the operation is subtraction, and the root power isn’t the same:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165225.image2.png\" alt=\"image2.png\" width=\"86\" height=\"38\" /></li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Multiplication and division</b> can be performed if the root powers are the same.</p>\r\n<p class=\"child-para\">Examples:</p>\r\n<p class=\"child-para\">These radicals can be combined because the operation is multiplication, and the root power is the same:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165226.image3.png\" alt=\"image3.png\" width=\"144\" height=\"38\" />\r\n<p class=\"child-para\">These radicals can be combined because the operation is division, and the root power is the same:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165227.image4.png\" alt=\"image4.png\" width=\"144\" height=\"38\" />\r\n<p class=\"child-para\">These radicals cannot be combined because the operation is division, and the root power isn’t the same:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165228.image5.png\" alt=\"image5.png\" width=\"88\" height=\"38\" /></li>\r\n</ul>\r\n<h2 id=\"tab2\" >Converting square roots to exponents</h2>\r\nFinding square roots and converting them to exponents is a relatively common operation in algebra. Square roots, which use the radical symbol, are nonbinary operations — operations which involve just one number — that ask you, “What number times itself gives you this number under the radical?” To convert the square root to an exponent, you use a fraction in the power to indicate that this stands for a root or a radical.\r\n\r\nWhen you find square roots, the symbol for that operation is a radical, which looks like this:\r\n<img src=\"https://www.dummies.com/wp-content/uploads/203683.image0.png\" alt=\"image0.png\" width=\"38\" height=\"38\" />\r\n\r\nWhen changing from radical form to fractional exponents, remember these basic forms:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">The <i>n</i>th root of <i>a</i> can be written as a fractional exponent with <i>a</i> raised to the reciprocal of that power.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/203684.image1.png\" alt=\"image1.png\" width=\"57\" height=\"33\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">When the <i>n</i>th root of</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/203685.image2.png\" alt=\"image2.png\" width=\"32\" height=\"32\" />\r\n<p class=\"first-para\">is taken, it’s raised to the 1/<i>n</i> power. When a power is raised to another power, you multiply the powers together, and so the <i>m</i> (otherwise written as <i>m</i>/1) and the 1/<i>n</i> are multiplied together.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/203686.image3.png\" alt=\"image3.png\" width=\"102\" height=\"50\" /></li>\r\n</ul>\r\nUse fractions in the powers to indicate that the expression stands for a root or a radical.\r\n\r\nHere are some examples of changing radical forms to fractional exponents:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/203687.image4.png\" alt=\"image4.png\" width=\"352\" height=\"286\" />\r\nWhen raising a power to a power, you multiply the exponents, but the bases have to be the same.\r\n\r\nBecause raising a power to a power means that you multiply exponents (as long as the bases are the same), you can simplify the following expressions:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/203688.image5.png\" alt=\"image5.png\" width=\"544\" height=\"106\" />\r\n\r\nLeave the exponent as 9/4. Don’t write it as a mixed number.\r\n\r\nThe following example can’t be combined because the bases are not the same:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/203689.image6.png\" alt=\"image6.png\" width=\"176\" height=\"50\" />","description":"<h2 id=\"tab1\" >Combining square roots</h2>\r\nSquare roots, which use the radical symbol, are nonbinary operations — operations which involve just one number — that ask you, “What number times itself gives you this number under the radical?” Finding square roots is a relatively common operation in algebra, but working with and combining the roots isn’t always so clear.\r\n\r\nExpressions with radicals can be multiplied or divided as long as the root power or value under the radical is the same. Expressions with radicals cannot be added or subtracted unless both the root power and the value under the radical are the same.\r\n<p class=\"Remember\">When you find square roots, the symbol for that operation is called a radical. The root power refers to the number outside and to the upper left of the radical. If there is no number, you assume that the root power is 2.</p>\r\nWhen it comes to combining radicals, there are just a couple of simple rules to remember:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Addition and subtraction</b> can be performed if the root power <i>and</i><b><i> </i></b>the value under the radical are the same.</p>\r\n<p class=\"child-para\">Examples:</p>\r\n<p class=\"child-para\">These radicals cannot be combined because the operation is addition, and the value under the radical is not the same:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165223.image0.png\" alt=\"image0.png\" width=\"88\" height=\"38\" />\r\n<p class=\"child-para\">These radicals can be combined because the root power and the numbers under the radical are the same:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165224.image1.png\" alt=\"image1.png\" width=\"176\" height=\"38\" />\r\n<p class=\"child-para\">These radicals cannot be combined because the operation is subtraction, and the root power isn’t the same:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165225.image2.png\" alt=\"image2.png\" width=\"86\" height=\"38\" /></li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Multiplication and division</b> can be performed if the root powers are the same.</p>\r\n<p class=\"child-para\">Examples:</p>\r\n<p class=\"child-para\">These radicals can be combined because the operation is multiplication, and the root power is the same:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165226.image3.png\" alt=\"image3.png\" width=\"144\" height=\"38\" />\r\n<p class=\"child-para\">These radicals can be combined because the operation is division, and the root power is the same:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165227.image4.png\" alt=\"image4.png\" width=\"144\" height=\"38\" />\r\n<p class=\"child-para\">These radicals cannot be combined because the operation is division, and the root power isn’t the same:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165228.image5.png\" alt=\"image5.png\" width=\"88\" height=\"38\" /></li>\r\n</ul>\r\n<h2 id=\"tab2\" >Converting square roots to exponents</h2>\r\nFinding square roots and converting them to exponents is a relatively common operation in algebra. Square roots, which use the radical symbol, are nonbinary operations — operations which involve just one number — that ask you, “What number times itself gives you this number under the radical?” To convert the square root to an exponent, you use a fraction in the power to indicate that this stands for a root or a radical.\r\n\r\nWhen you find square roots, the symbol for that operation is a radical, which looks like this:\r\n<img src=\"https://www.dummies.com/wp-content/uploads/203683.image0.png\" alt=\"image0.png\" width=\"38\" height=\"38\" />\r\n\r\nWhen changing from radical form to fractional exponents, remember these basic forms:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">The <i>n</i>th root of <i>a</i> can be written as a fractional exponent with <i>a</i> raised to the reciprocal of that power.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/203684.image1.png\" alt=\"image1.png\" width=\"57\" height=\"33\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">When the <i>n</i>th root of</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/203685.image2.png\" alt=\"image2.png\" width=\"32\" height=\"32\" />\r\n<p class=\"first-para\">is taken, it’s raised to the 1/<i>n</i> power. When a power is raised to another power, you multiply the powers together, and so the <i>m</i> (otherwise written as <i>m</i>/1) and the 1/<i>n</i> are multiplied together.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/203686.image3.png\" alt=\"image3.png\" width=\"102\" height=\"50\" /></li>\r\n</ul>\r\nUse fractions in the powers to indicate that the expression stands for a root or a radical.\r\n\r\nHere are some examples of changing radical forms to fractional exponents:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/203687.image4.png\" alt=\"image4.png\" width=\"352\" height=\"286\" />\r\nWhen raising a power to a power, you multiply the exponents, but the bases have to be the same.\r\n\r\nBecause raising a power to a power means that you multiply exponents (as long as the bases are the same), you can simplify the following expressions:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/203688.image5.png\" alt=\"image5.png\" width=\"544\" height=\"106\" />\r\n\r\nLeave the exponent as 9/4. Don’t write it as a mixed number.\r\n\r\nThe following example can’t be combined because the bases are not the same:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/203689.image6.png\" alt=\"image6.png\" width=\"176\" height=\"50\" />","blurb":"","authors":[],"primaryCategoryTaxonomy":{"categoryId":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Combining square roots","target":"#tab1"},{"label":"Converting square roots to exponents","target":"#tab2"}],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice 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Algebra Articles

Master algebra and algebra II with quick breakdowns of key math skills, plus practice problem sets and other cool stuff.

Articles From Algebra

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Algebra Algebra Workbook For Dummies Cheat Sheet

Cheat Sheet / Updated 04-28-2022

Formulas, patterns, and procedures used for simplifying expressions and solving equations are basic to algebra. Use the equations, shortcuts, and formulas you find for quick reference. This Cheat Sheet offers basic information and short explanations (and some words of advice on traps to avoid).

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Algebra Algebra II: 1001 Practice Problems For Dummies Cheat Sheet

Cheat Sheet / Updated 04-19-2022

The best way to figure out how the different algebraic rules work and interact with one another is to practice with lots of problems. And Algebra II requires lots of practice. So be prepared to solve equations and systems, graph lines, tackle functions, and so much more.

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Algebra Algebra II Workbook For Dummies Cheat Sheet

Cheat Sheet / Updated 04-12-2022

Learning some algebraic rules for various exponents, radicals, laws, binomials, formulas, and equations will help you successfully study and solve problems in an Algebra II course. You should also be able to recognize formulas to find slope, slope-intercept, distance, and midpoint (which are formulas from geometry) to help you through Algebra II.

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Algebra Algebra I: 1001 Practice Problems For Dummies Cheat Sheet

Cheat Sheet / Updated 03-28-2022

Algebra problems are easier to solve when you know the rules and formulas. Unlike other subjects where you can just read or listen and absorb the information sufficiently, math takes practice. The only way to figure out how the different algebraic rules work and interact with one another is to get into the problems and get your hands dirty, so to speak. Be prepared to practice on lots of different problems.

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Algebra Linear Algebra For Dummies Cheat Sheet

Cheat Sheet / Updated 03-14-2022

To study and solve linear algebra equations successfully, you need to know common numerical values of trig functions, what elements determine a vector space, basic algebraic properties, and general commands using graphing calculators to solve linear algebra problems.

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Algebra Algebra I For Dummies Cheat Sheet

Cheat Sheet / Updated 02-24-2022

Algebra problems are easier to solve when you know the rules and formulas. Memorizing key algebra formulas will speed up your work, too. And if you know the rules of divisibility and the order of operations, you'll be able to solve algebra problems without getting stressed.

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Algebra Algebra II For Dummies Cheat Sheet

Cheat Sheet / Updated 01-21-2022

Algebra is all about formulas, equations, and graphs. You need algebraic equations for multiplying binomials, dealing with radicals, finding the sum of sequences, and graphing the intersections of cones and planes. You also get to deal with logarithms, you lucky Algebra II user!

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Algebra Convert a Circle Equation to the Standard Form

Article / Updated 12-21-2021

When the equation of a circle appears in the standard form, it provides you with all you need to know about the circle: its center and radius. With these two bits of information, you can sketch the graph of the circle. The equation x2 + y2 + 6x – 4y – 3 = 0, for example, is the equation of a circle. You can change this equation to the standard form by completing the square for each of the variables. Just follow these steps: Change the order of the terms so that the x's and y's are grouped together and the constant appears on the other side of the equal sign. Leave a space after the groupings for the numbers that you need to add: x2 + 6x _____ + y2 – 4y _____ = 3 Complete the square for each variable, adding the number that creates perfect square trinomials. In the case of the x's, you add 9, and with the y's, you add 4. Don't forget to also add 9 and 4 to the right: x2 + 6x + 9 + y2 – 4y + 4 = 3 + 9 + 4 When it's simplified, you have x2 + 6x + 9 + y2 – 4y + 4 = 16 Factor each perfect square trinomial. The standard form for the equation of this circle is (x + 3)2 + (y – 2)2 = 16. The circle has its center at the point (–3, 2) and has a radius of 4 (the square root of 16). To sketch this circle, you locate the point (–3, 2) and then count 4 units up, down, left, and right; sketch in a circle that includes those points. The figure below shows you the way. With the center, radius, and a compass, you too can sketch this circle.

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Algebra How to Add and Subtract with Powers

Article / Updated 07-14-2021

To add or subtract with powers, both the variables and the exponents of the variables must be the same. You perform the required operations on the coefficients, leaving the variable and exponent as they are. When adding or subtracting with powers, the terms that combine always have exactly the same variables with exactly the same powers. These rules are true for multiplying and dividing exponents as well. Example 1: x + x + x = 3x Because the variables are the same (x) and the powers are the same (there are no exponents, so the exponents must be 1), you can add the variables. Example 2: Because the variables are the same (x) and the powers are the same (2), you can perform the required operations on the variables. Example 3: Although the variables are the same (x), the powers are not the same (1, 2, 3, and 4). You can't simplify these terms because only the variables are the same, and both the variables and the powers need to be the same. Example 4: Sometimes not all of the variables and powers will be the same — you may encounter a problem that has several groups of variables and powers that are not the same. In this case, you only add or subtract terms whose variables and powers are the same. (Notice that the exponents are listed in order from highest to lowest. This is a common practice to make answers easy to compare.)

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Algebra How to Combine Square Roots and Convert Square Roots to Exponents

Article / Updated 07-14-2021

Combining square roots Square roots, which use the radical symbol, are nonbinary operations — operations which involve just one number — that ask you, “What number times itself gives you this number under the radical?” Finding square roots is a relatively common operation in algebra, but working with and combining the roots isn’t always so clear. Expressions with radicals can be multiplied or divided as long as the root power or value under the radical is the same. Expressions with radicals cannot be added or subtracted unless both the root power and the value under the radical are the same. When you find square roots, the symbol for that operation is called a radical. The root power refers to the number outside and to the upper left of the radical. If there is no number, you assume that the root power is 2. When it comes to combining radicals, there are just a couple of simple rules to remember: Addition and subtraction can be performed if the root power and the value under the radical are the same. Examples: These radicals cannot be combined because the operation is addition, and the value under the radical is not the same: These radicals can be combined because the root power and the numbers under the radical are the same: These radicals cannot be combined because the operation is subtraction, and the root power isn’t the same: Multiplication and division can be performed if the root powers are the same. Examples: These radicals can be combined because the operation is multiplication, and the root power is the same: These radicals can be combined because the operation is division, and the root power is the same: These radicals cannot be combined because the operation is division, and the root power isn’t the same: Converting square roots to exponents Finding square roots and converting them to exponents is a relatively common operation in algebra. Square roots, which use the radical symbol, are nonbinary operations — operations which involve just one number — that ask you, “What number times itself gives you this number under the radical?” To convert the square root to an exponent, you use a fraction in the power to indicate that this stands for a root or a radical. When you find square roots, the symbol for that operation is a radical, which looks like this: When changing from radical form to fractional exponents, remember these basic forms: The nth root of a can be written as a fractional exponent with a raised to the reciprocal of that power. When the nth root of is taken, it’s raised to the 1/n power. When a power is raised to another power, you multiply the powers together, and so the m (otherwise written as m/1) and the 1/n are multiplied together. Use fractions in the powers to indicate that the expression stands for a root or a radical. Here are some examples of changing radical forms to fractional exponents: When raising a power to a power, you multiply the exponents, but the bases have to be the same. Because raising a power to a power means that you multiply exponents (as long as the bases are the same), you can simplify the following expressions: Leave the exponent as 9/4. Don’t write it as a mixed number. The following example can’t be combined because the bases are not the same:

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