General form of $\sqrt{a - b} - \sqrt{a + b}$? 
General form of $\sqrt{a - b} - \sqrt{a + b}$?

What I would do is: let 
$x = \sqrt{a - b} - \sqrt{a + b}$
$x^2 = 2a - 2\sqrt{a^2 - b^2}$
Then since $a + b > a - b$
$x = -\sqrt{2a - 2\sqrt{a^2 - b^2}}$
This is particularly useful in solving problem with nested square roots (like, let $a = 11, b = 2\sqrt{10}$
I am wondering, is there any other derivation and MORE WANTED: any simpler formula? 

You want to calculate $u-v$ for $u$ and $v$ such that $u^2=a-b$ and $v^2=a+b$. Let us assume that $a>b$, i.e., $u,v>0$.
Then you have $u^2v^2=(a-b)(a+b)=a^2-b^2$ which leaves you with
$$uv=\sqrt{a^2-b^2}.$$
Then you have
$$(u-v)^2=u^2+v^2-2uv=2a-2\sqrt{a^2+b^2}$$
and, knowing that $u<v$ and $u-v<0$, you get
$$u-v = \sqrt{2a-2\sqrt{a^2+b^2}}.$$
Similarly, you can use
$$(u+v)^2=u^2+v^2+2uv = 2a+2\sqrt{a^2+b^2}$$
to calculate
$$u+v=\sqrt{2a+2\sqrt{a^2+b^2}}.$$
