Question

Rút gọn biểu thức:

a)\(\frac{\left(2+\sqrt{3}\right)\sqrt{2-\sqrt{3}}}{\sqrt{2}+\sqrt{3}}\)

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

Answer 1

a) \(\frac{\left(2+\sqrt{3}\right)\sqrt{2-\sqrt{3}}}{\sqrt{2}+\sqrt{3}}\)

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

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

\(=\frac{\sqrt{2+\sqrt{3}}\cdot\sqrt{4-3}}{\sqrt{2}+\sqrt{3}}\)

\(=\frac{\sqrt{2+\sqrt{3}}}{\sqrt{2}+\sqrt{3}}\)

b) Xét \(\left(\sqrt{3+\sqrt{5}}+\sqrt{3-\sqrt{5}}\right)^2\)

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

\(=6+2\sqrt{9-5}\)

\(=6+2\cdot2\)

\(=10\)