# GRE Math : How to find a ratio of square roots

## Example Questions

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

Which of the following is equivalent to the ratio of  to ?

Explanation:

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

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:

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

What is  if  ?

Explanation:

Note that

This changes the initial equation to

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

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

Given the equation , solve for .

Explanation:

In order to solve for , the equation  must be written such that each set of exponents shares the same base:

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

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

Which of the following is equal to ?

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:

By simplifying, we get .

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

Which of the following has the same ratio as ?

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:

If we redistribute the negative, then the answer becomes

.

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

What is the ratio of  expressed in  form?