Question

Simplify:

If $\frac{7}{\sqrt{17}-2\sqrt{3}}-\frac{3}{\sqrt{17}+2\sqrt{3}}$ is $\frac{4\sqrt{17}+20\sqrt{3}}{m}$, then $m$ is ?

Answer 1

Given $\frac{7}{\sqrt{17}-2\sqrt{3}}-\frac{3}{\sqrt{17}+2\sqrt{3}}$
Rationalize the denominators with the respective factors,
$\frac{7}{\sqrt{17}-2\sqrt{3}}\times \frac{\sqrt{17}+2\sqrt{3}}{\sqrt{17}+2\sqrt{3}}-\frac{3}{\sqrt{17}+2\sqrt{3}}\times \frac{\sqrt{17}-2\sqrt{3}}{\sqrt{17}-2\sqrt{3}}$
$=\frac{7(\sqrt{17}+2\sqrt{3})-3(\sqrt{17}-2\sqrt{3})}{17-4\times 3}$
$=\frac{4\sqrt{17}+20\sqrt{3}}{5}$

$\therefore$ Value of $m=5$