Need help with simplifying a radical expression

Question

I need help with simplifying this radical expression:

$\sqrt{(5+2\sqrt{6})}(49-20\sqrt{6})(9\sqrt{3}+11\sqrt{2})$.

Answer 1

Noting that $$\sqrt{5+2\sqrt 6}=\sqrt{(2+3)+2\sqrt{2\times 3}}=\sqrt{(\sqrt 2+\sqrt 3)^2}=\sqrt 2+\sqrt 3,$$ we have $$\begin{align}(\sqrt 2+\sqrt 3)(49-20\sqrt 6)&=49\sqrt 2-20\cdot 2\sqrt 3+49\sqrt 3-20\cdot 3\sqrt 2\\&=-11\sqrt 2+9\sqrt 3.\end{align}$$

Hence, we have $$\begin{align}(-11\sqrt 2+9\sqrt 3)(9\sqrt 3+11\sqrt 2)&=(9\sqrt 3-11\sqrt 2)(9\sqrt 3+11\sqrt 2)\\&=(9\sqrt 3)^2-(11\sqrt 2)^2\\&=243-242\\&=1.\end{align}$$

Hence, $$\sqrt{(5+2\sqrt 6)}\ (49-20\sqrt 6)(9\sqrt 3+11\sqrt 2)=1.$$