Question

Simplify:

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

Answer 1

To simplify: $\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}+\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$

LCM of $\sqrt{5}-\sqrt{3}$ and $\sqrt{5}+\sqrt{3}$

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

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

$=5–3$

$=2$

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

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

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

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

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

$=\frac{16}{2}$

$=8$