How to find a ratio of square roots - GRE Quantitative Reasoning

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 .

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

.

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

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

Final answer is .

Note that

This changes the initial equation to

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

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 :

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:

Carefully FOIL your denominator and distribute your numerator:

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

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 .

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

.

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

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

Final answer is .

Note that

This changes the initial equation to

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

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 :

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:

Carefully FOIL your denominator and distribute your numerator:

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

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 .

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

.

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

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

Final answer is .

Note that

This changes the initial equation to

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

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 :

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:

Carefully FOIL your denominator and distribute your numerator:

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

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 .