2011 AMC 10A Problem 16

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.