McGraw-Hill Education GRE 2019

Note that when determining the square root of a number, the answer will

always be positive. Even though (−4)^2 = 16, −4 is not a solution for x in

the question.

In the preceding example, 16 is a perfect square. A perfect square is any

number whose square root is an integer. For example, 9 is a perfect square because

its square root is 3, but 15 is not a perfect square, since its square root is 3.87....

Multiplying and Dividing Roots

If you are asked to simply evaluate a square root, you can of course use your

on-screen calculator. But many root questions will require you to instead

manipulate the root. For example: √ 32 × √ 2 = ?.

When multiplying square roots, you can combine all the terms underneath one

square root. Thus

√32 × √ 2 = √32 × 2 = √ 64 = 8

When dividing square roots, you can combine all the terms underneath one square

root:

√32

√2 = √

32

2 = √^16 = 4

Note that these rules also work in reverse:

√200 = √100 × 2 = √ 100 × √ 2 = 10√ 2

Simplifying Roots

Simplifying a perfect square root is straightforward and can always be done on

your calculator. But what if, after going through a question, you arrived at an

answer of √ 32? If you looked at the choices, √ 32 would not appear in any of them.

Why? Because √ 32 is not simplified. To simplify it, you have to take any perfect

squares out of the radical. You would simplify √ 32 in the following way:

√32 = √16 × 2 = √ 16 × √ 2 = 4√ 2

4√ 2 is the simplified form of √ 32. Generally, when you are trying to simplify a

square root, you should break it up into any known perfect squares and remove

those perfect squares from the square root.

Si mpl i f y. √ 150

SOLUTION: First, rewrite 150 as 25 × 6. Thus √ 150 = √25 × 6. Now, break up the

square root: √25 × 6 = √ 25 × √ 6. Finally, simplify any perfect squares. √ 25 = 5,

so the answer is 5√ 6.

CHAPTER 11 ■ ALGEBRA 275

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