How to simplify $\frac{4 + 2\sqrt6}{\sqrt{5 + 2\sqrt{6}}}$? I was tackling through an olympiad practice book when I saw one of these problems:

If  $x = 5 + 2\sqrt6$, evaluate $\Large{x  \ - \ 1 \over\sqrt{x}}$?

The answer written is $2\sqrt2$, but I can't figure my way out through the manipulations. I just know that I have the following:$${4+ 2\sqrt6} \over {\sqrt{5 + 2\sqrt6}}$$

$$\frac{x-2}{\sqrt x}=\frac{4+ 2\sqrt6}{\sqrt{5 + 2\sqrt6}}=\frac{4+ 2\sqrt6}{\sqrt{5 + 2\sqrt6}}\cdot\frac{\sqrt{5-2\sqrt 6}}{\sqrt{5-2\sqrt 6}}=\left(4+2\sqrt 6\right)\sqrt{5-2\sqrt 6}\Longrightarrow$$
$$(x-2)^2=x(40+16\sqrt 6)(5-2\sqrt 6)=x(8)\Longrightarrow $$
$$x^2-12x+4=0\ldots...$$
Take it from here
