Question

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

Answer 1

$\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}+\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}=\frac{(\sqrt{5}-\sqrt{3}{)}^2+(\sqrt{5}+\sqrt{3}{)}^2}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})}\left[\text{By taking L.C.M}\right]\because (a-b{)}^2=a^2+b^2-2ab\text{and}(a+b{)}^2=a^2+b^2+2ab=\frac{(\sqrt{5}{)}^2+(\sqrt{3}{)}^2-2\left(\sqrt{5}\right)\left(\sqrt{3}\right)+(\sqrt{5}{)}^2+(\sqrt{3}{)}^2+2\left(\sqrt{5}\right)\left(\sqrt{3}\right)}{(\sqrt{5}{)}^2-(\sqrt{3}{)}^2}$

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

$=\frac{10+6}{5-3}=\frac{16}{8}=8$