Question

Simplify:$\left(\frac{\sqrt{5}-1}{\sqrt{5}+1}+\frac{\sqrt{5}+1}{\sqrt{5}-1}\right)$

Answer 1

$$\begin{gathered} \frac{\sqrt{5}-1}{\sqrt{5}+1}+\frac{\sqrt{5}+1}{\sqrt{5}-1} \\ \mathrm{Taking}\mathrm{the}\mathrm{LCM}\mathrm{of}\mathrm{both}\mathrm{the}\mathrm{fractions}\mathrm{and}\mathrm{adding}\mathrm{them},\mathrm{we}\mathrm{get}: \\ \frac{{\left(\sqrt{5}-1\right)}^2+{\left(\sqrt{5}+1\right)}^2}{\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)} \\ =\frac{5+1-2\sqrt{5}+5+1+2\sqrt{5}}{{\left(\sqrt{5}\right)}^2-{\left(1\right)}^2} \\ =\frac{12}{5-1} \\ =\frac{12}{4} \\ =3 \end{gathered}$$