GRE Math : How to find a ratio of square roots

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Example Question #61 : Arithmetic

Which of the following is equivalent to the ratio of $\sqrt{10}$ to $\sqrt{2} - 1$ ?

Possible Answers:

$\frac{\sqrt{10}}{\sqrt{3}}$

$\frac{\sqrt{5}(2+\sqrt{2})}{3}$

$\sqrt{5}(2+\sqrt{2})$

$\frac{2\sqrt{7}}{\sqrt{10}}$

$\frac{2\sqrt{5}}{\sqrt{3}}$

Correct answer:

$\sqrt{5}(2+\sqrt{2})$

Explanation:

At first, this problem seems rather easy. You merely need to divide these two values to get:

$\frac{\sqrt{10}}{\sqrt{2}-1}$

However, there are no answers that look like this! When this happens, you should consider rationalizing the denominator to eliminate the square root. This is a little more difficult than normal problems like this (ones that contain only the radical). However, if you complete a difference of squares in the denominator, you will be well on your way to having the right answer:

$\frac{\sqrt{10}}{\sqrt{2}-1} = \frac{\sqrt{10}}{\sqrt{2}-1} \cdot \frac{\sqrt{2}+1}{\sqrt{2}+1}$

Carefully FOIL your denominator and distribute your numerator:

$\frac{\sqrt{10}}{\sqrt{2}-1} \cdot \frac{\sqrt{2}+1}{\sqrt{2}+1} = \frac{\sqrt{20}+\sqrt{10}}{2-1}$

Look to your answers for an idea for factoring your numerator:

$\frac{\sqrt{20}+\sqrt{10}}{1} = 2\sqrt{5}+\sqrt{10} = \sqrt{5}(2+\sqrt{2})$

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Example Question #62 : Arithmetic

What is if $4^8=16^{x+1}$ ?

Possible Answers:

Correct answer:

Explanation:

Note that $4^8=4^{2(4)}=16^4$

This changes the initial equation to

$16^4=16^{x+1}$

For this equation to be valid, the exponents must be equal:

4=x+1

3=x

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Example Question #63 : Arithmetic

Given the equation $4^{6x+12}=8^{8x+24}$ , solve for .

Possible Answers:

-3.6

3.6

2.4

Correct answer:

Explanation:

In order to solve for , the equation $4^{6x+12}=8^{8x+24}$ must be written such that each set of exponents shares the same base:

$4^{6x+12}=8^{8x+24}$

$4^{3(2x+4)}=8^{2(4x+12)}$

$64^{2x+4}=64^{4x+12}$

Which like bases, it's now just a matter of solving for :

2x+4=4x+12

2x=-8

x=-4

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Example Question #1 : How To Find A Ratio Of Square Roots

Which of the following is equal to $\frac{\sqrt{2}+1}{\sqrt{2}-1}$ ?

Possible Answers:

$3+2\sqrt{2}$

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

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

Correct answer:

$3+2\sqrt{2}$

Explanation:

To find an equivalent, just multiply the top and bottom by the conjugate of the denominator.

Conjugate is the square root expression found in the denominator but with opposite sign.

So:

$\frac{\sqrt{2}+1}{\sqrt{2}-1}\cdot \frac{\sqrt{2}+1}{\sqrt{2}+1}=\frac{2+2\sqrt{2}+1}{2-1}$

By simplifying, we get $3+2\sqrt{2}$ .

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Example Question #2 : How To Find A Ratio Of Square Roots

Which of the following has the same ratio as $\frac{\sqrt{2}}{\sqrt{3}-2}$ ?

Possible Answers:

$\frac{6-2\sqrt{2}}{1}$

$\frac{6+2\sqrt{2}}{1}$

$\frac{2\sqrt{2}+\sqrt{6}}{1}$

$\frac{\sqrt{6}-2\sqrt{2}}{1}$

$\frac{-2\sqrt{2}-\sqrt{6}}{1}$

Correct answer:

$\frac{-2\sqrt{2}-\sqrt{6}}{1}$

Explanation:

Since in all the answer choices have an integer in the denominator, we should multiply top and bottom by the conjugate of the denominator which is the square root expression with opposite sign.

So:

$\frac{\sqrt{2}}{\sqrt{3}-2}\cdot \frac{\sqrt{3}+2}{\sqrt{3}+2}=\frac{\sqrt{6}+2\sqrt{2}}{-1}$

If we redistribute the negative, then the answer becomes

$\frac{-\sqrt{6}-2\sqrt{2}}{1}$ .

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Example Question #3 : How To Find A Ratio Of Square Roots

What is the ratio of $\frac{\frac{\sqrt{2}}{\sqrt{7}}}{\frac{\sqrt{3}}{\sqrt{5}}}$ expressed in a:b form?

Possible Answers:

$\sqrt{10}:\sqrt{21}$

$\sqrt{21}:\sqrt{10}$

10:21

$\sqrt{14}:\sqrt{15}$

$\sqrt{6}:\sqrt{35}$

Correct answer:

$\sqrt{10}:\sqrt{21}$

Explanation:

To get into a:b form, multiply the fraction by bottom denominator's reciprocal.

$\frac{\frac{\sqrt{2}}{\sqrt{7}}}{\frac{\sqrt{3}}{\sqrt{5}}}\cdot \frac{\frac{\sqrt{5}}{\sqrt{3}}}{\frac{\sqrt{5}}{\sqrt{3}}}=\frac{\sqrt{10}}{\sqrt{21}}$

The is the numerator of the fraction and is the denominator.

Final answer is $\sqrt{10}:\sqrt{21}$ .

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GRE Math : How to find a ratio of square roots

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #61 : Arithmetic

Example Question #62 : Arithmetic

Example Question #63 : Arithmetic

Example Question #1 : How To Find A Ratio Of Square Roots

Example Question #2 : How To Find A Ratio Of Square Roots

Example Question #3 : How To Find A Ratio Of Square Roots

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