Question

Simplify: $\left(\frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}\right)$.

Answer 1

$$\begin{gathered} \frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}} \\ =\frac{{\left(4+\sqrt{5}\right)}^2+{\left(4-\sqrt{5}\right)}^2}{\left(4-\sqrt{5}\right)\left(4+\sqrt{5}\right)} \\ =\frac{4^2+{\left(\sqrt{5}\right)}^2+2\times 4\times \sqrt{5}+4^2+{\left(\sqrt{5}\right)}^2-2\times 4\times \sqrt{5}}{16-5} \\ =\frac{16+5+8\sqrt{5}+16+5-8\sqrt{5}}{16-5} \\ =\frac{42}{11} \end{gathered}$$