Question

Rút gọn: \(\sqrt{6+2\sqrt{5}}+\sqrt{\frac{2}{7+3\sqrt{5}}}\)

Answer 1

\(\sqrt{6+2\sqrt{5}}+\sqrt{\frac{2}{7+3\sqrt{5}}}\)

\(=\sqrt{1^2+2\sqrt{5}+\left(\sqrt{5}\right)^2}+\sqrt{\frac{2\left(7-3\sqrt{5}\right)}{49-9\cdot5}}\)

\(=\sqrt{\left(\sqrt{5}+1\right)^2}+\sqrt{\frac{14-6\sqrt{5}}{4}}\)

\(=\sqrt{5}+1+\sqrt{\frac{3^2-2\cdot3\cdot\sqrt{5}+\left(\sqrt{5}\right)^2}{4}}\)

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

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