Question

\(\dfrac{\sqrt{5}+3}{\sqrt{2}+\sqrt{3+\sqrt{5}}}\) + \(\dfrac{3-\sqrt{5}}{\sqrt{2}-\sqrt{3-\sqrt{5}}}\)

Answer 1

\(\dfrac{3+\sqrt{5}}{\sqrt{2}+\sqrt{3+\sqrt{5}}}+\dfrac{3-\sqrt{5}}{\sqrt{2}-\sqrt{3-\sqrt{5}}}=\dfrac{\sqrt{2}\left(3+\sqrt{5}\right)}{2+\sqrt{6+2\sqrt{5}}}+\dfrac{\sqrt{2}\left(3-\sqrt{5}\right)}{2-\sqrt{6-2\sqrt{5}}}\)

\(=\dfrac{\sqrt{2}\left(3+\sqrt{5}\right)}{2+\sqrt{\left(\sqrt{5}+1\right)^2}}+\dfrac{\sqrt{2}\left(3-\sqrt{5}\right)}{2-\sqrt{\left(\sqrt{5}-1\right)^2}}\)

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