The TASC Math exam will likely contain some questions where you'll have to confront radicals. Don't worry—it's not as scary as it sounds. You can just think of radicals as the "opposite" operation of applying exponents.

The symbol


is called a radical or root. Unless otherwise indicated, you're looking for the


or principal root when it's an even index (the number outside the root symbol). Otherwise, you must consider both the positive and negative versions:


Note that if the index of the radical is not explicitly written, then you assume it's the square root, so the index is 2.

It's important to remember perfect squares. You should probably know up to 12 at least, if not 15.


To simplify a radical expression, follow these steps:

  1. Make a factor tree (go to the primes, meaning you can't break the number down anymore).
  2. Find pairs (square root), triples (cubed root), and so on.
  3. Write pairs outside the radical (write only once) and leave everything else inside (think of the pairs as friends going out to play).
  4. Find the products of what you "pulled out" and left behind.
Take this example:


Practice questions

  1. Solve for x:
  2. Simplify

Answers and explanations

  1. The correct answer is Choice (B). To solve this type of problem, you can substitute each of the choices in for x and solve or you can rewrite the original problem to be in the same base. Because the right side of the equation is a fraction, you know that its exponent is going to be negative: 3x = 27–1. Now you can rewrite 27 in base 3: 3x = (33)–1 = 3–3. This means that x = –3, Choice (B).
  2. The correct answer is Choice (C). When combining radicals, you first have to check that the radicands (the expression under the radical symbol) are the same. Because they are in this problem, you add the two coefficients together and leave the radicands alone, which results in Choice (C).

About This Article

This article is from the book:

About the book authors:

Stuart Donnelly, PhD, earned his doctorate in mathematics from Oxford University at the age of 25. Since then, he has established successful tutoring services in both Hong Kong and the United States and is considered by leading educators to be one of the most experienced and qualified private tutors in the country. Nicole Hersey, PhD, is a lecturer at the University of Rhode Island, with a dual appointment to the School of Education and the Department of Mathematics. Ron Olson, MA, is an NBCT-certified teacher in Social Studies who teaches AP Government, Civics, and Contemporary World Problems at Clover Park High School in Lakewood, WA. In addition to his 35 years of teaching experience, he works as an AP US History workshop consultant for The College Board and has been the advisor for National Honor Society at his high school. Shannon Reed, MA, MFA, is a visiting lecturer at the University of Pittsburgh, where she teaches composition, creative writing, and business writing.

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