Question

Prove that
$\frac{1}{3\sqrt{2}-2\sqrt{3}}=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}$

Answer 1

L.H.S$=\frac{1}{3\sqrt{2}-2\sqrt{3}}$

$=\frac{1}{3\sqrt{2}-2\sqrt{3}}\times \frac{3\sqrt{2}+2\sqrt{3}}{3\sqrt{2}+2\sqrt{3}}$

$=\frac{3\sqrt{2}+2\sqrt{3}}{{\left(3\sqrt{2}\right)}^2-{\left(2\sqrt{3}\right)}^2}$

$=\frac{3\sqrt{2}+2\sqrt{3}}{18-12}$

$=\frac{3\sqrt{2}+2\sqrt{3}}{6}$

$=\frac{\sqrt{2}}{2}\times \frac{\sqrt{2}}{\sqrt{2}}+\frac{\sqrt{3}}{3}\times \frac{\sqrt{3}}{\sqrt{3}}$

$=\frac{2}{2\sqrt{2}}+\frac{3}{3\sqrt{3}}$

$=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}=$R.H.S

Hence proved.