GRE Math : Simplifying Square Roots

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Example Questions

Example Question #22 : Arithmetic

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

Possible Answers:

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

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

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

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

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

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

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Example Question #1 : How To Simplify Square Roots

What is $\sqrt{50}$ ?

Possible Answers:

$10\sqrt{2}$

$2\sqrt{5}$

$5\sqrt{2}$

Correct answer:

$5\sqrt{2}$

Explanation:

We know that 25 is a factor of 50. The square root of 25 is 5. That leaves $\sqrt{2}$ which can not be simplified further.

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Example Question #1 : Simplifying Square Roots

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

Possible Answers:

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

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

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

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

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

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:

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

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

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

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

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Example Question #2 : Simplifying Square Roots

Which of the following is the most simplified form of:

$\sqrt{468}$

Possible Answers:

$2\sqrt{117}$

$6\sqrt{13}$

$17\sqrt{2}$

$4\sqrt{29}$

$\sqrt{468}$

Correct answer:

$6\sqrt{13}$

Explanation:

First find all of the prime factors of 468

$468=6\ast78=6\ast6\ast13=2\ast3\ast2\ast3\ast13$

So $\sqrt{468}=\sqrt{2\ast2\ast3\ast3\ast13}=2\ast3\sqrt{13}=6\sqrt{13}$

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Example Question #3 : Simplifying Square Roots

What is $\sqrt{432}$ equal to?

Possible Answers:

$144\sqrt{3}$

$6\sqrt{3}$

$12\sqrt{3}$

$6\sqrt{4}$

$12\sqrt{12}$

Correct answer:

$12\sqrt{3}$

Explanation:

$\sqrt{432}$

1. We know that 432=(144)(3) , which we can separate under the square root:

$\sqrt{144\cdot 3}$

2. 144 can be taken out since it is a perfect square: $12\cdot12=144$ . This leaves us with:

$12\sqrt{3}$

This cannot be simplified any further.

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GRE Math : Simplifying Square Roots

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #22 : Arithmetic

Example Question #1 : How To Simplify Square Roots

Example Question #1 : Simplifying Square Roots

Example Question #2 : Simplifying Square Roots

Example Question #3 : Simplifying Square Roots

All GRE Math Resources

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