$ \sqrt{2-\sqrt{2}} $ simplified So I have a (nested?) square root $ \sqrt{2-\sqrt{2}} $. I know that $ \sqrt{2-\sqrt{3}} = \frac{\sqrt{6}-\sqrt{2}}{2} $. I know how to turn the simplified version into the complex one, but not vice versa. What is $ \sqrt{2-\sqrt{2}} $ simplified in this fashion and what are the steps?

A general way to solve such problems:
$$\sqrt{2-\sqrt{2}}=a$$
then
$a^2=2-\sqrt{2}\rightarrow(2-a^2)^2=2\rightarrow 4-4a^2+a^4=2\rightarrow a^4-4a^2+2=0$
Let $b=a^2$
then
Solve $$b^2-4b+2=0$$
As @Martin suggested, this is a way but the conslusion is not right. The right way should be
$$\sqrt{a+b\sqrt{c}}=\sqrt{d}+\sqrt{e}$$ and the following arithmetic similar to above.
