Question

Rút gọn

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

Answer 1

\(A=\frac{2\left(3+\sqrt{5}\right)}{4+\sqrt{6+2\sqrt{5}}}+\frac{2\left(3-\sqrt{5}\right)}{4-\sqrt{6-2\sqrt{5}}}=\frac{6+2\sqrt{5}}{4+\sqrt{\left(\sqrt{5}+1\right)^2}}+\frac{6-2\sqrt{5}}{4-\sqrt{\left(\sqrt{5}-1\right)^2}}\)

\(=\frac{6+2\sqrt{5}}{4+\sqrt{5}+1}+\frac{6-2\sqrt{5}}{4-\left(\sqrt{5}-1\right)}=\frac{6+2\sqrt{5}}{5+\sqrt{5}}+\frac{6-2\sqrt{5}}{5-\sqrt{5}}\)

\(=\frac{\left(6+2\sqrt{5}\right)\left(5-\sqrt{5}\right)+\left(6-2\sqrt{5}\right)\left(5+\sqrt{5}\right)}{\left(5-\sqrt{5}\right)\left(5+\sqrt{5}\right)}=\frac{40}{20}=2\)