# How to Distribute Variables

Distributing variables over the terms in an algebraic expression involves multiplication rules and the rules for exponents. When different variables are multiplied together, you can write them side by side without using any multiplication symbols. If the same variable is multiplied as part of the distribution, then you add the exponents.

The exponent rule says that when multiplying exponents with the same base, you add the exponents:

This sample problem illustrates the exponent rule for multiplying terms with the same base.

Distribute the term outside the parentheses over the terms within.

Add the exponents.

Complete the distribution.

The next example contains negative powers and fractional powers — the rules for exponents remain the same with negative and fractional exponents.

*Example:*

Distribute the term outside the parentheses over the terms within the parentheses.

Add the exponents.

Complete the distribution.

The exponent zero means the value of the expression is one:

You combine exponents with different signs by using the rules for adding and subtracting signed numbers. Fractional exponents are combined after finding common denominators. Exponents that are improper fractions are left in that form.

Try going through many of the situations that could arise when distributing, such as distributing both a number and a variable, distributing various powers of more than one variable, distributing negatives, rewriting negative exponents as fractional terms, distributing fractional powers, and distributing radicals. This touches on just about anything you’d probably come across.

Combine the variables by using the rules for exponents.

*Example 1:*** **Distribute 5*x* through the expression

Multiply each term by 5x.

Then multiply the numbers and the variables in each term.

*Example 2: *Combine the variables with the same base using the rules for exponents. The signs of the results follow the rules for multiplying signed numbers.

–6*y*(5*xy* – 4*x* – 3*y* + 2)

Multiply each term by -6*y*.

–6*y*(5*xy*) – 6*y*(–4*x*) – 6*y*(–3*y*) – 6*y*(2)

Do the multiplication in each term.

Algebra Glossary

absolute value

An operation that tells you how far a number is from zero.

Algebra Glossary

additive inverse

The number with the same numerical part but the opposite sign (plus or minus) of a given number. If zero is the sum of two numbers, then these two numbers are additive inverses of one another.

Algebra Glossary

associative property

A characteristic of addition and multiplication that allows the grouping of terms to change without affecting the result.

Algebra Glossary

base

A value multiplied repeatedly in an exponential expression.

Algebra Glossary

binary operation

A process requiring two values to produce a third value.

Algebra Glossary

binomial

Two terms separated by addition or subtraction.

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braces

A grouping symbol that looks like this: { }.

Algebra Glossary

brackets

A grouping symbol that looks like this: [ ].

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coefficient

A number multiplied by a variable. The coefficient precedes the variable.

Algebra Glossary

combinations

A method of counting that tells how many ways a designated number of objects can be selected from a given set.

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common denominator

The same value on the bottom of more than one fraction.

Algebra Glossary

commutative property

A characteristic of addition and multiplication that allows the order of the values in an operation to be changed without affecting the result.

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composite number

A whole number larger than 1 that isn’t prime.

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constant

A variable or number that never changes in value in an expression.

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cube; perfect cube

The third power of a number; the result of multiplying a number by itself three times.

Algebra Glossary

cube root

A number that you can multiply by itself three times to get a given number. For example, the cube root of 8 is 2, because 2 times 2 times 2 equals 8.

Algebra Glossary

cubic

An adjective that describes an expression in which the highest power is three.

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decimal

A fraction with an unwritten denominator of 10 indicated by the decimal point.

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degree of an expression

The highest power occurring in an expression.

Algebra Glossary

denominator

The bottom number in a fraction.

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difference

The result of subtraction.

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digit

Numerals from zero through nine, so called because they were originally counted on the fingers.

Algebra Glossary

distributive property

A characteristic of multiplication and addition that allows for the multiplication of each individual term in a grouped series by a term outside of the grouping without changing the value of the expression.

Algebra Glossary

dividend

A number to be divided by another number.

Algebra Glossary

divisible

A number that can be divided by another number with no remainder.

Algebra Glossary

double root

A solution that appears twice when solving an equation because the related factor appears twice in the factored form. For example, in (*y *– 2)(*y* – 2) = 0, *y* = 2 is a double root or solution. In (*y *– 2)(*y *– 2)(*x* + 3) = 0, *y* = 2 is still a double root or solution.

Algebra Glossary

equation

A mathematical statement with an equal sign showing that two values are equal.

Algebra Glossary

equivalent fractions

Fractions equal to one another, even though they may have different denominators.

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even number

A number that can be divided by 2.

Algebra Glossary

exponent; power

A value in smaller type found above and to the right of the base that indicates the number of times the base is multiplied by itself.

Algebra Glossary

expression

Any combination of values (variables, numbers, and/or constants) and operations that can be used to show how things belong together and compare to one another.

Algebra Glossary

factor

1. (noun) Any of the values involved in a multiplication problem that when multiplied together produces a result. 2. (verb) To rewrite an algebraic expression as a product.

Algebra Glossary

factorial

An operation that multiplies a whole number by every counting number smaller than the whole number.

Algebra Glossary

FOIL

1. An acronym for first, outer, inner, and last — which indicates the order in which two terms (one from each of two binomials) are multiplied together. 2. The process of multiplying two binomials.

Algebra Glossary

fraction

Any quantity expressed as a numerator (the value above the fraction line) and a denominator (the value below the fraction line).

Algebra Glossary

fraction line

A line separating the numerator from the denominator in a fraction.

Algebra Glossary

greatest common factor; GCF

The largest possible value that evenly divides each term of an expression containing two or more terms.

Algebra Glossary

grouping symbol

Parentheses, brackets, and braces that can affect the order of operations. Terms and operations within the grouping symbol take precedence.

Algebra Glossary

groups

Terms and operations within grouping symbols.

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improper fraction

A fraction whose numerator is larger than the denominator.

Algebra Glossary

inequality

A relationship between two unequal values.

Algebra Glossary

infinite

Without end; uncountable, as in a repeating decimal.

Algebra Glossary

integer

A positive or negative whole number or zero; numbers starting with zero and going up or down in increments of one.

Algebra Glossary

irrational number

A number with no fractional equivalent whose decimal never repeats or terminates.

Algebra Glossary

linear

An adjective describing an expression or equation in which the highest power of any variable is one. Constants can be higher powers.

Algebra Glossary

lowest terms

The status of a fraction whose numerator and denominator cannot be divided evenly by the same number.

Algebra Glossary

mixed number

An improper fraction written as a whole number alongside a fraction.

Algebra Glossary

monomial

An expression with only one term.

Algebra Glossary

multiple

A number evenly divisible by a specific factor.

Algebra Glossary

multiplication property of zero; MPZ

A rule stating if the product of two numbers is zero, then one of the numbers must be zero.

Algebra Glossary

multiplicative inverse

Also known as a reciprocal, one of two numbers whose product is one. The reciprocal of a number is that particular number in the denominator of a fraction with a value of one in the numerator.

Algebra Glossary

multiplier

An integer multiplied with both the numerator and denominator of a fraction.

Algebra Glossary

natural numbers; counting numbers

Values starting with one and increasing by one.

Algebra Glossary

negative number

Any quantity that is less than zero; it is usually preceded by a minus sign.

Algebra Glossary

negative reciprocals

Two numbers, one positive and one negative, whose product is –1.

Algebra Glossary

nested

A condition where two or more grouping symbols are inside one another.

Algebra Glossary

number system

A set of numbers.

Algebra Glossary

numerator

The top number in a fraction.

Algebra Glossary

odd number

A whole number that is not evenly divided by 2.

Algebra Glossary

operation

A mathematical process, such as addition, subtraction, multiplication, and division, performed on one or more quantities.

Algebra Glossary

percent

Fractions with a denominator of 100. The percentage is the numerator of the fraction, indicating the number of parts out of 100.

Algebra Glossary

permutation

A counting method that determines the number of ordered arrangements there are when a certain number of objects are selected from a given set.

Algebra Glossary

polynomial

An expression with one or more terms.

Algebra Glossary

positive number

Any quantity greater than zero.

Algebra Glossary

power

A value of an exponent indicating the number of times the base is multiplied by itself.

Algebra Glossary

power of zero

A rule of exponents, where any number raised to the power of zero equals one as long as the base number isn’t zero.

Algebra Glossary

prime factorization

The process of finding the prime numbers that, when multiplied together, produce a given composite number.

Algebra Glossary

prime number

A whole number larger than one that can be divided evenly only by itself and one.

Algebra Glossary

principal square root

A positive number that when multiplied by itself produces a given positive number.

Algebra Glossary

product

The result of multiplication.

Algebra Glossary

proper fraction

A fraction whose value is less than one; thus, the numerator is always smaller than the denominator.

Algebra Glossary

proportion

An equation showing that two ratios are equal to one another.

Algebra Glossary

quadratic

Also known as *second degree**,* an expression or equation in which the highest power is 2.

Algebra Glossary

quotient

The result of division.

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radical

The symbol for the operation to find a square root.

Algebra Glossary

raised to the power of

Multiplying a base number the number of times indicated by the exponent.

Algebra Glossary

rational number

A quantity, positive or negative, that can be written as a fraction; its decimal equivalent terminates or repeats.

Algebra Glossary

real number

Any rational or irrational number.

Algebra Glossary

reciprocal; multiplicative inverse

1. Two numbers whose product is always one. 2. Either one of the two numbers in a reciprocal.

Algebra Glossary

reduce

To divide out a common factor of the numerator and denominator of a fraction, leaving an equivalent fraction.

Algebra Glossary

relatively prime

The condition of two numbers that have no factors in common other than the number one.

Algebra Glossary

remainder

A value that is left over when one number is divided by another.

Algebra Glossary

repeating decimal; recurring decimal

A decimal in which, beyond a certain point, a digit or set of digits repeats indefinitely.

Algebra Glossary

root

A value that, multiplied by itself a number of times, results in the value or number wanted.

Algebra Glossary

rounding

Approximating a value to the nearest digit or decimal place.

Algebra Glossary

scientific notation

A standard way of writing very large and very small numbers as the product of two values — a number between 1 and 9 and a power of 10. Scientific notation follows the form *N* × 10*a** *where *N *is a number from 1 up to 10, but not 10 itself, and *a *is an integer (positive or negative number).

Algebra Glossary

sign

A symbol indicating whether a value is positive (+) or negative (−).

Algebra Glossary

simple fraction

A fraction in which both the numerator and the denominator are whole numbers.

Algebra Glossary

simplify

To combine all that can be combined in an expression or equation, and put it in its most easily understandable form.

Algebra Glossary

solution of equation

Value(s) of a variable that make an equation a true statement.

Algebra Glossary

solve

Find the answer or the number that a variable stands for.

Algebra Glossary

square; perfect square

1. The product of another number times itself. 2. A value with an exponent of 2.

Algebra Glossary

square root

A value resulting when a given value is multiplied by itself.

Algebra Glossary

substitution

A method of replacing a value with its equivalent.

Algebra Glossary

sum

The result of addition.

Algebra Glossary

symmetric property

A characteristic of equations that allows for the exchange of the value(s) on one side of the equal sign with the value(s) on the other side (quantities on the right go to the left; quantities on the left go to the right) without changing the truth of the equation: If *x* = *y,* then *y* = *x.*

Algebra Glossary

synthetic division

A short-cut division process in which only the coefficients of the terms in an expression are used. The answer is obtained by multiplying and adding.

Algebra Glossary

term

A group of number(s) and/or variable(s) connected to one another by multiplication or division and separated from other terms by addition or subtraction.

Algebra Glossary

trinomial

An expression with three terms. Each term is separated from the others by addition or subtraction.

Algebra Glossary

unFOIL

To factor a trinomial into two binomials.

Algebra Glossary

value

A numeric equivalent or worth of an expression or variable.

Algebra Glossary

variable

A letter representing an unknown number or what you’re solving for in an algebra problem.

Algebra Glossary

whole numbers

All natural numbers plus zero.