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

Question

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?

Answer 1

"Simpler" is dependent on the use made of the formula. For approximate calculations, the form $$\frac{-2b}{\sqrt{a - b} \; + \; \sqrt{a + b}}$$ (obtained by multiplying numerator and denominator by $\sqrt{a - b} \; + \; \sqrt{a + b}\,)$ is better when $|b|$ is very small in comparison to $|a|.$