Question

Rationalise the denominator of the following:
$\frac{3\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$

Answer 1

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

Dividing numerator and denominator by $\sqrt{5}+\sqrt{3}$

$=\frac{3\sqrt{5}+\sqrt{3}(\sqrt{5}+\sqrt{3})}{\sqrt{5}-\sqrt{3}(\sqrt{5}+\sqrt{3})}$

$=\frac{3\sqrt{5}\sqrt{5}+3\sqrt{5}\sqrt{3}+\sqrt{3}\sqrt{5}+\sqrt{3}\sqrt{3}}{\sqrt{5}\sqrt{5}-\sqrt{3}\sqrt{3}}$

$=\frac{15+3\sqrt{15}+\sqrt{15}+3}{5-3}$

$=\frac{18+4\sqrt{15}}{2}$

$=9+2\sqrt{15}$