Simplify $\sqrt{6-\sqrt{20}}$ My first try was to set the whole expression equal to $a$ and square both sides. $$\sqrt{6-\sqrt{20}}=a \Longleftrightarrow a^2=6-\sqrt{20}=6-\sqrt{4\cdot5}=6-2\sqrt{5}.$$
Multiplying by conjugate I get $$a^2=\frac{(6-2\sqrt{5})(6+2\sqrt{5})}{6+2\sqrt{5}}=\frac{16}{2+\sqrt{5}}.$$
But I still end up with an ugly radical expression.

There's actually a general formula for these kinds of expressions. Namely$$\sqrt{X\pm Y}=\sqrt{\frac {X+\sqrt{X^2-Y^2}}2}\pm\sqrt{\frac {X-\sqrt{X^2-Y^2}}2}$$
Where $X,Y$ are real numbers. Simply substituting $X=6$ and $Y=\sqrt{20}$ gives the proper denesting. The proof of this is quite simple. Assume that$$X\pm Y=\left(\sqrt A\pm \sqrt B\right)^2$$and expand via binomial theorem. Collecting terms, you will end up with two equations from which you can solve for $A$ and $B$ in terms of $X$ and $Y$.
