Question

\(\sqrt{\frac{3-\sqrt{5}}{3+\sqrt{5}}}+\sqrt{\frac{3+\sqrt{5}}{3-\sqrt{5}}}\)=?

Answer 1

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

\(=\frac{\sqrt{\left(3-\sqrt{5}\right)^2}}{\sqrt{9-5}}+\frac{\sqrt{\left(3+\sqrt{5}\right)^2}}{\sqrt{9-5}}=\frac{3-\sqrt{5}}{2}+\frac{3+\sqrt{5}}{2}\left(\text{ vì }3>\sqrt{5}\right)\)

\(=\frac{6}{2}=3\)