Question

Thực hiện phép tính:

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

Answer 1

\(=\frac{\sqrt{2}\left(3+\sqrt{5}\right)}{\sqrt{6-2\sqrt{5}}}+\frac{\sqrt{2}\left(3-\sqrt{5}\right)}{\sqrt{6+2\sqrt{5}}}=\frac{\sqrt{2}\left(3+\sqrt{5}\right)}{\sqrt{\left(\sqrt{5}-1\right)^2}}+\frac{\sqrt{2}\left(3-\sqrt{5}\right)}{\sqrt{\left(\sqrt{5}+1\right)^2}}\)

\(=\frac{\sqrt{2}\left(3+\sqrt{5}\right)}{\sqrt{5}-1}+\frac{\sqrt{2}\left(3-\sqrt{5}\right)}{\sqrt{5}+1}=\frac{\sqrt{2}\left(3+\sqrt{5}\right)\left(\sqrt{5}+1\right)}{4}+\frac{\sqrt{2}\left(3-\sqrt{5}\right)\left(\sqrt{5}-1\right)}{4}\)

\(=\frac{\sqrt{2}\left(8+4\sqrt{5}\right)}{4}+\frac{\sqrt{2}\left(4\sqrt{5}-8\right)}{4}=2\sqrt{10}\)