Question

Simplify:

$\frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}$

Answer 1

$\frac{(4+\sqrt{5})\times (4+\sqrt{5})}{(4-\sqrt{5})\times (4+\sqrt{5})}+\frac{(4-\sqrt{5})\times (4-\sqrt{5})}{(4+\sqrt{5})\times (4-\sqrt{5})}$

$\Rightarrow \frac{(4+\sqrt{5}{)}^2}{(16-5)}+\frac{(4-\sqrt{5}{)}^2}{(16-5)}$ [ Applying $(a^2-b^2)=(a+b)(a-b)$]

$\Rightarrow \frac{(16+5+8\sqrt{5}+16+5-8\sqrt{5})}{11}$

$\Rightarrow \frac{42}{11}$