Simplfying $\sqrt{31-8\sqrt{15}}+\sqrt{31+8\sqrt{15}}$ I am trying to simplify the expression:
$\sqrt{31-8\sqrt{15}}+\sqrt{31+8\sqrt{15}}$
I tried to square the expression but I can't do that because it is not an equation so I got stuck. Can someone please give me some pointers on how to proceed?

The general way to evaluate these kinds of expressions is to try to factor the contents of the square root into a perfect square.
One can try to find the factorization by letting $31 + 8\sqrt{15} = (a + b\sqrt{15})^2 = (a^2 + 15b^2) + 2ab\sqrt{15}$. We then test some possible values of $a$ and $b$. In this case, we see that $a = 4$ and $b = 1$ works. So, we can deduce that :
$$\begin{align}(4 + \sqrt{15})^2 &= 16 + 15 + 8 \sqrt{15} \\
&= 31 + 8 \sqrt{15}\end{align}$$
Similarly, we can deduce that 
$$\begin{align}(4 - \sqrt{15})^2 &= 16 + 15 - 8 \sqrt{15} \\
&= 31 - 8 \sqrt{15}\end{align}$$
Hence,
$$\begin{align}\sqrt{31-8\sqrt{15}}+\sqrt{31+8\sqrt{15}} &= \sqrt{(4 - \sqrt{15})^2} + \sqrt{(4 + \sqrt{15})^2} \\
&= 4 - \sqrt{15} + 4 + \sqrt{15} \\ 
&= 8\end{align}$$
Of course, you can alternatively go ahead to obtain the solution by squaring the original expression, but it'll be rather tedious if you are working with bigger numbers.
