Question

Simplify the following: $\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$.

Answer 1

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

$=\frac{(\sqrt{5}+\sqrt{3})(\sqrt{5}+\sqrt{3})+(\sqrt{5}-\sqrt{3})(\sqrt{5}-\sqrt{3})}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})}$ ...(Rationalising the denominator)

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

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

$=\frac{5+3+2\sqrt{15}+5+3-2\sqrt{15}}{2}$

$=\frac{16}{2}=8$.