Question

If $x=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ and $y=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$ then find $(x+y)$

Answer 1

$x+y=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}=\frac{{(\sqrt{5}+\sqrt{3})}^2+{(\sqrt{5}-\sqrt{3})}^2}{{\sqrt{5}}^2-{\sqrt{3}}^2}=\frac{{\sqrt{5}}^2+{\sqrt{3}}^2+2\times \sqrt{5}\times \sqrt{3}+{\sqrt{5}}^2+{\sqrt{3}}^2-2\times \sqrt{5}\times \sqrt{3}}{5-3}=\frac{16}{2}=8$