GRE Quantitative Reasoning - How to subtract square roots

Quantity A: $\sqrt{140}-\sqrt{315}$

Quantity B: $\sqrt{132}-\sqrt{297}$

Which of the following is true?

Quantity B is larger.

Quantity A is larger.

The two quantities are equal.

The relationship cannot be determined from the information provided.

Explanation

As always for comparison questions, work on getting your quantities explicit. In order to do this in this case, you must factor the roots.

Quantity A

$\sqrt{140}-\sqrt{315}$

$\sqrt{4*35}-\sqrt{9*35}$

This can be factored into:

$2\sqrt{35}-3\sqrt{35}=-\sqrt{35}$

Quantity B

$\sqrt{132}-\sqrt{297}$

$\sqrt{4*33}-\sqrt{9*33}$

This can be factored into:

$2\sqrt{33}-3\sqrt{33}=-\sqrt{33}$

This makes the comparison very easy! Since is a larger number than , we know that $-\sqrt{35}$ is a smaller value than $-\sqrt{33}$ . Therefore, Quantity B is larger.

Solve for .

Note, $x\ge0$ :

$5\sqrt{x}=\sqrt{30}+4\sqrt{x}$

$\sqrt{30}$

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

$\sqrt{15}$

Explanation

Begin by subtracting $4\sqrt{x}$ from both sides:

$5\sqrt{x}-4\sqrt{x}=\sqrt{30}$

This simplifies to:

$\sqrt{x}=\sqrt{30}$

Now, while you could square both sides, you also know that merely needs to be the same as the value under the other radical. This means that it is —simply and quickly! (Always work to save time on the exam!)