<s>[INST] Simplify $\frac{1}{\sqrt{-1+\sqrt{2}}}$ In an exercise I got as solution $\frac{1}{\sqrt{-1+\sqrt{2}}}$, it now holds that $\frac{1}{\sqrt{-1+\sqrt{2}}} = \sqrt{1+\sqrt{2}}$. But I really don't see how you can manipulate the left hand side to become the right hand side (quite shameful as a mathematician, I must admit). 
Does anyone know how to do it?
 [/INST] $\frac{1}{\sqrt{\sqrt{2} - 1}} \cdot \frac{\sqrt{\sqrt{2} + 1}}{\sqrt{\sqrt{2} + 1}}$
= $\frac{\sqrt{\sqrt{2} + 1}}{\sqrt{(\sqrt{2})^2 - 1}}$
= $\frac{\sqrt{\sqrt{2} + 1}}{\sqrt{2 - 1}}$
= $\sqrt{\sqrt{2} + 1}$
= $\sqrt{1 + \sqrt{2}}$
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