Q:

Explain the process of solving this nested squareroot problem

I am in class 10 and this is an olympiad question so I am having a problem doing this. The innermost one I have evaluated to be $\sqrt{15}- \sqrt2$. But the rest I don't know how to do.
$\sqrt { 2 + \sqrt { 3 } - \sqrt { 4 + \sqrt { 5 } + \sqrt { 17 - 4 \sqrt { 15 } } } }$

A:

Well actually, the inner most evaluates to $(\sqrt{12}-\sqrt{5})$ and not what you said. You can solve such questions as:-
$$a^2+b^2=17$$
$$2ab=4\sqrt{15}$$
Solve these equations to get the desired result (Also, note that b is negative, but you can change these signs later knowing that there's $-2ab$ over there). It'll then evaluate to:-
$$\sqrt{2+\sqrt{3}-\sqrt{4+\sqrt{5}+\sqrt{12}-\sqrt{5}}}$$
$$=\sqrt{2+\sqrt{3}-\sqrt{4+2\sqrt{3}}}$$
$$=\sqrt{2+\sqrt{3}-\sqrt{3}-1}$$
$$=1.$$

