Question

Rút gọn biểu thức:

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

Answer 1

h) Ta có: \(\sqrt{\dfrac{3+\sqrt{5}}{\sqrt{3-\sqrt{5}}}}+\sqrt{\dfrac{3-\sqrt{5}}{\sqrt{3+\sqrt{5}}}}\)

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

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

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

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