Question

Rationalize the denominator: $\frac{\sqrt{7}}{\sqrt{5}+\sqrt{3}}$.

Answer 1

$\frac{\sqrt{7}}{\sqrt{5}+\sqrt{3}}$

$=\frac{\sqrt{7}\times (\sqrt{5}-\sqrt{3})}{(\sqrt{5}+\sqrt{3})\times (\sqrt{5}-\sqrt{3})}$

$=\frac{\sqrt{7}(\sqrt{5}-\sqrt{3})}{(\sqrt{5}{)}^2-(\sqrt{3}{)}^2}$

$=\frac{\sqrt{7}(\sqrt{5}-\sqrt{3})}{5-3}$

$=\frac{\sqrt{7}}{2}(\sqrt{5}-\sqrt{3})$

$\therefore \frac{\sqrt{7}}{\sqrt{5}+\sqrt{3}}=\frac{\sqrt{7}}{2}(\sqrt{5}-\sqrt{3})$