How to evaluate $\sqrt{5+2\sqrt{6}}$ + $\sqrt{8-2\sqrt{15}}$?

HINT:

What is $\displaystyle (\sqrt{2} + \sqrt{3})^2 $ and $\displaystyle (\sqrt{5} - \sqrt{3})^2 $?

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The basic 'trick', so to say, behind such questions is to identify that the surd can be expressed as a square. For example, consider your first surd $\sqrt{5+2\sqrt{6}}$. Here, there's a $2\cdot\sqrt{\text{something}}$. Now, if you see, the factors of that $\sqrt{\text{something}}$ are $\sqrt2$ and $\sqrt3$. A quick check shows that the sum of their squares give $5$. So, this is of the form $a^2 + b^2 + 2ab = (a+b)^2$

There is a relevant discussion in Dummit and Foote with respect to biquadratic extensions: $\sqrt{a+\sqrt{b}}=\sqrt{m}+\sqrt{n}$ if and only if $a^2-b$ is a perfect square.