<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] Set up an equation like:
$$
u = \sqrt{31 -8 \sqrt{15}} + \sqrt{31 + 8 \sqrt{15}}
$$
Square to get it down to:
$$
u^2 = 31 - 8 \sqrt{15} + 31 + 8 \sqrt{15} + 2 \sqrt{31^2 - 8^2 \cdot 15}
    = 62 + 2 \cdot 1 = 64
$$
Thus you have $u = \pm 8$. The original expression is clearly positive, so:
$$
\sqrt{31 -8 \sqrt{15}} + \sqrt{31 + 8 \sqrt{15}} = 8
$$
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