$ \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?

this is a general way to do it:
$\sqrt{2-\sqrt{2}}=\sqrt x+\sqrt y $
then $2-\sqrt 2=2 \sqrt{xy}+x+y$
and then make the rational parts equal to the rational parts as such:
$2=x+y$
now we know $y=2-x$. substitute this into the irrational part:
$-\sqrt 2 =2\sqrt{(x)}\sqrt{(2-x)}$ square both sides to get
$2=4(x)(2-x) \rightarrow 2=8x-4x^2$
However the soulutions to this quadratic equation are $1-\frac{1}{\sqrt{2}}$ and $1+\frac{1}{\sqrt{2}}$ 

so     $ \sqrt{ 2-\sqrt{2}}=\sqrt{1-\frac{1}{\sqrt{2}}} +\sqrt{1+\frac{1}{\sqrt{2}}}$

In other words there is no way to denest your radical.
