GRE Math : How to simplify square roots

Study concepts, example questions & explanations for GRE Math

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Example Questions

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Example Question #1 : How To Simplify Square Roots

Simplify the following: (√(6) + √(3)) / √(3)

Possible Answers:

1

3√(2)

√(3)

None of the other answers

√(2) + 1

Correct answer:

√(2) + 1

Explanation:

Begin by multiplying top and bottom by √(3):

(√(18) + √(9)) / 3

Note the following:

√(9) = 3

√(18) = √(9 * 2) = √(9) * √(2) = 3 * √(2)

Therefore, the numerator is: 3 * √(2) + 3.  Factor out the common 3: 3 * (√(2) + 1)

Rewrite the whole fraction:

(3 * (√(2) + 1)) / 3

Simplfy by dividing cancelling the 3 common to numerator and denominator: √(2) + 1

Example Question #3 : How To Simplify Square Roots

what is 

√0.0000490

Possible Answers:

49

0.00007

7

0.007

0.07

Correct answer:

0.007

Explanation:

easiest way to simplify: turn into scientific notation

√0.0000490= √4.9 X 10-5

finding the square root of an even exponent is easy, and 49 is  a perfect square, so we can write out an improper scientific notation:

√4.9 X 10-5√49 X 10-6

√49 = 7; √10-6 = 10-3 this is equivalent to raising 10-6 to the 1/2 power, in which case all that needs to be done is multiply the two exponents: 7 X 10-3= 0.007

Example Question #2 : How To Simplify Square Roots

Simplify:

Possible Answers:

Correct answer:

Explanation:

In order to take the square root, divide 576 by 2.

Example Question #3 : How To Simplify Square Roots

Simplify (\frac{16}{81})^{1/4}.

Possible Answers:

\frac{2}{3}

\frac{4}{9}

\frac{4}{81}

\frac{2}{81}

\frac{8}{81}

Correct answer:

\frac{2}{3}

Explanation:

(\frac{16}{81})^{1/4}

\frac{16^{1/4}}{81^{1/4}}

\frac{(2\cdot 2\cdot 2\cdot 2)^{1/4}}{(3\cdot 3\cdot 3\cdot 3)^{1/4}}

\frac{2}{3}

Example Question #4 : How To Simplify Square Roots

Simplfy the following radical .

Possible Answers:

Correct answer:

Explanation:

You can rewrite the equation as .

This simplifies to .

Example Question #5 : How To Simplify Square Roots

Which of the following is equal to  ?

Possible Answers:

Correct answer:

Explanation:

√75 can be broken down to √25 * √3. Which simplifies to 5√3.

Example Question #6 : How To Simplify Square Roots

Simplify \sqrt{a^{3}b^{4}c^{5}}.

Possible Answers:

a^{2}bc\sqrt{bc}

a^{2}b^{2}c^{2}\sqrt{bc}

a^{2}bc^{2}\sqrt{ac}

ab^{2}c^{2}\sqrt{ac}

a^{2}b^{2}c\sqrt{ab}

Correct answer:

ab^{2}c^{2}\sqrt{ac}

Explanation:

Rewrite what is under the radical in terms of perfect squares:

x^{2}=x\cdot x

x^{4}=x^{2}\cdot x^{2}

x^{6}=x^{3}\cdot x^{3}

Therefore, \sqrt{a^{3}b^{4}c^{5}}= \sqrt{a^{2}a^{1}b^{4}c^{4}c^{1}}=ab^{2}c^{2}\sqrt{ac}.

Example Question #7 : How To Simplify Square Roots

What is ?

Possible Answers:

Correct answer:

Explanation:

We know that 25 is a factor of 50. The square root of 25 is 5. That leaves  which can not be simplified further.

Example Question #46 : Basic Squaring / Square Roots

Which of the following is equivalent to \frac{x + \sqrt{3}}{3x + \sqrt{2}}?

Possible Answers:

\frac{3x^{2} + \sqrt{6}}{3x - 2}

\frac{3x^{2} + 3x\sqrt{2} + x\sqrt{3} +\sqrt{6}}{9x^{2} - 2}

\frac{3x^{2} - \sqrt{6}}{9x^{2} + 2}

\frac{4x + \sqrt{5}}{3x + 2}

\frac{3x^{2} -x \sqrt{2} + 3x\sqrt{3} - \sqrt{6}}{9x^{2} - 2}

Correct answer:

\frac{3x^{2} -x \sqrt{2} + 3x\sqrt{3} - \sqrt{6}}{9x^{2} - 2}

Explanation:

Multiply by the conjugate and the use the formula for the difference of two squares:

\frac{x + \sqrt{3}}{3x + \sqrt{2}}

\frac{x + \sqrt{3}}{3x + \sqrt{2}}\cdot \frac{3x - \sqrt{2}}{3x - \sqrt{2}}

\frac{3x^{2} -x \sqrt{2} + 3x\sqrt{3} - \sqrt{6}}{(3x)^{2} - (\sqrt{2})^{2}} 

\frac{3x^{2} -x \sqrt{2} + 3x\sqrt{3} - \sqrt{6}}{9x^{2} - 2}

Example Question #47 : Basic Squaring / Square Roots

Which of the following is the most simplified form of:

 

Possible Answers:

Correct answer:

Explanation:

First find all of the prime factors of 

So 

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