<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] Here is another way of doing it.
If we set $a=31+8\sqrt {15}, b=31-8\sqrt {15}$ we have 
$a+b=62$ and 
$ab=31^2-15\cdot 64=31^2-30\cdot32=1$
We know that $ab$ will work out well, because we know that $(x+y)(x-y)=x^2-y^2$
Then we know that $a$ and $b$ are the roots of $x^2-62x+1=0$
We want the square roots $\alpha$ and $\beta$ of $a$ and $b$ so if we set $y^2=x$ we will create an equation which has these as roots - $y^4-62y^2+1=0$
We know that $ab=1$ because we computed that. When we take the square root we get that $\alpha\beta=1$.
We therefore look for a factorisation $(y^2+py+1)(y^2-py+1)=y^4-62y^2+1$ which gives us $2-p^2=-62$, or $p^2=64$.
We know that one of these factors will have roots $\alpha, \beta$ and the other will have roots $-\alpha, -\beta$ so the final step is to work out which of these gives us the answer.
I have glossed over a couple of things because this is homework - but the double square root coming from a quartic without any terms in $y$ or $y^3$ is closely related to the underlying field theory, and is worth noticing.
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