Question

rút gọn: \(\frac{1}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{1}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)

Answer 1

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

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

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

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

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

\(=\frac{\sqrt{2}\left(3-\sqrt{3}\right)+\sqrt{2}\left(3+\sqrt{3}\right)}{9-3}\)

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

\(=\frac{6\sqrt{2}}{6}=\sqrt{2}\)