1/(1 - √x) - 1/(1 + √x) + √x/(1 - x) | Indices - Simplicity

Simplify: $\dfrac{1}{1- \sqrt{x}} - \dfrac{1}{1+ \sqrt{x}} + \dfrac{\sqrt{x}}{1 - x}$

Solution:

$= \dfrac{1}{1 - \sqrt{x}} - \dfrac{1}{1 + \sqrt{x}} + \dfrac{\sqrt{x}}{1 - x}$

$= \dfrac{1(1 + \sqrt{x}) - 1(1 - \sqrt{x})}{(1+\sqrt{x})(1 - \sqrt{x})} + \dfrac{\sqrt{x}}{1 - x}$

$= \dfrac{1 + \sqrt{x} - 1 + \sqrt{x}}{1^2 - ( \sqrt{x})^2} + \dfrac{\sqrt{x}}{1 - x}$

$= \dfrac{2 \sqrt{x}}{1 - x} + \dfrac{ \sqrt{x}}{1 - x}$

$= \dfrac{2\sqrt{x} + \sqrt{x}}{1 -x}$

$= \dfrac{3 \sqrt{x} }{1 - x}$

Answer