Question: Rationalize the denominators and simplify: $\dfrac{\sqrt{a+b} - \sqrt{a-b}}{\sqrt{a+b} + \sqrt{a-b}}$


Solution:
Given,

$= \dfrac{\sqrt{a+b} - \sqrt{a-b}}{\sqrt{a+b} + \sqrt{a-b}}$

$= \dfrac{\sqrt{a+b} - \sqrt{a-b}}{\sqrt{a+b} + \sqrt{a-b}} × \dfrac{\sqrt{a+b} - \sqrt{a-b}}{\sqrt{a+b} - \sqrt{a-b}}$

$= \dfrac{(\sqrt{a+b} - \sqrt{a-b})^2}{(\sqrt{a+b})^2 - (\sqrt{a-b})^2}$

$= \dfrac{(\sqrt{a+b})^2 - 2×\sqrt{a+b}× \sqrt{a-b} + (\sqrt{a-b})^2}{a+b-(a-b)}$

$= \dfrac{a+b - 2 \sqrt{(a+b)(a-b)} + a-b}{a+b-a+b}$

$= \dfrac{2a - 2\sqrt{a²-b²}}{2b}$

$= \dfrac{2(a - \sqrt{a²-b²}}{2b}$

$= \dfrac{a - \sqrt{a² -b²}}{b}$
= Answer