Question

Simplify $\frac{\sqrt{5}}{\sqrt{5}+2}-\frac{\sqrt{3}}{\sqrt{5}-2}$

Answer 1

We have $\frac{\sqrt{5}}{\sqrt{5}+2}-\frac{\sqrt{3}}{\sqrt{5}-2}$ where R.F of $\sqrt{5}+2$ is $\sqrt{5}-2$ and R.F of $\sqrt{5}-2$ is $\sqrt{5}+2$
$\therefore \frac{\sqrt{5}}{\sqrt{5}+2}-\frac{\sqrt{3}}{\sqrt{5}-2}=\frac{\sqrt{5}}{\sqrt{5}+2}\times \frac{\sqrt{5}-2}{\sqrt{5}-2}-\frac{\sqrt{3}}{\sqrt{5}-2}\times \frac{\sqrt{5}+2}{\sqrt{5}+2}$
$=\frac{\sqrt{5}(\sqrt{5}-2)}{(\sqrt{5}{)}^2-2^2}-\frac{\sqrt{3}(\sqrt{5}+2)}{(\sqrt{5}{)}^2-2^2}$
$=\frac{5-2\sqrt{5}}{5-4}-\frac{(\sqrt{15}+2\sqrt{3})}{5-4}$
$=5-2\sqrt{5}-\sqrt{15}-2\sqrt{3}$
$\therefore \frac{\sqrt{5}}{\sqrt{5}+2}-\frac{\sqrt{3}}{\sqrt{5}-2}=5-2\sqrt{5}-\sqrt{15}-2\sqrt{3}$