2011 AMC 10A Problem 16

Question

Which of the following is equal to

\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}?

3\sqrt2

2\sqrt6

\dfrac{7\sqrt2}{2}

3\sqrt3

6

Square the whole expression

The product

(9-6\sqrt2)(9+6\sqrt2)

is small

Since we have square roots, we can try to change the inside of each radical to be a perfect square.

Note that we can rewrite the expression as

\begin{aligned} & \sqrt{6 - 6\sqrt2 + 3} \\ &{}+ \sqrt{6 + 6\sqrt2 + 3}. \end{aligned}

Factoring and simplifying gives us

\sqrt{(\sqrt6 - \sqrt3)^2} + \sqrt{(\sqrt6 + \sqrt3)^2}

= \sqrt6 - \sqrt3 + \sqrt6 + \sqrt3 = 2\sqrt6.

Thus, B is the correct answer.

Problem 16 in other years