Question

Rationalize the denominator of the following:
$\frac{4\sqrt{3}+5\sqrt{2}}{\sqrt{48}+\sqrt{18}}$.

Answer 1

$\frac{4\sqrt{3}+5\sqrt{2}}{\sqrt{48}+\sqrt{18}}=\frac{4\sqrt{3}+5\sqrt{2}}{4\sqrt{3}+3\sqrt{2}}$

$=\frac{4\sqrt{3}+5\sqrt{2}}{4\sqrt{3}+3\sqrt{2}}\times \frac{4\sqrt{3}-3\sqrt{2}}{4\sqrt{3}-3\sqrt{2}}$

$=\frac{48-12\sqrt{6}+20\sqrt{6}-30}{(4\sqrt{3}{)}^2-(3\sqrt{2}{)}^2}$

$=\frac{18+8\sqrt{6}}{48-18}=\frac{18+8\sqrt{6}}{30}=\frac{9+4\sqrt{6}}{15}$.