Simplifying expression $\sqrt{6+2\sqrt{5}} - \sqrt{6-2\sqrt{5}}.$ I'm stuck. Squaring it will change it's value. Is there any general method of simplifying expressions of the form $$\sqrt{a+b}-\sqrt{a-b} = c \ \ ?$$

If I were you I would tackle the individual square roots to find$$\sqrt{6+2\sqrt{5}} = \sqrt{5}+1$$ $$\sqrt{6-2\sqrt{5}} = \sqrt{5}-1$$ and see that the difference is $2$. 
Note: It is not always possible to denest a square root like this, but if our goal is to try to find integers $a$, $b$ such that $\sqrt{6+2\sqrt{5}}=\sqrt{a}+\sqrt{b}$, we can try to solve $a$, $b$ by $$6+2\sqrt{5} = a+b+2\sqrt{ab}\implies a+b=6, \ \ ab=5\implies (a,b)=(1,5) \ \text{or} \ (5,1)$$
