<s>[INST] 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?
 [/INST] 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.
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