Ta có: \(\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)
\(=\frac{2\sqrt{2}+\sqrt{6}}{2+\sqrt{4+2\sqrt{3}}}+\frac{2\sqrt{2}-\sqrt{6}}{2-\sqrt{4-2\sqrt{3}}}\)
\(=\frac{2\sqrt{2}+\sqrt{6}}{2+\sqrt{\left(\sqrt{3}+1\right)^2}}+\frac{2\sqrt{2}-\sqrt{6}}{2-\sqrt{\left(\sqrt{3}-1\right)^2}}\)
\(=\frac{2\sqrt{2}+\sqrt{6}}{2+\left|\sqrt{3}+1\right|}+\frac{2\sqrt{2}-\sqrt{6}}{2-\left|\sqrt{3}-1\right|}\)
\(=\frac{2\sqrt{2}+\sqrt{6}}{2+\sqrt{3}+1}+\frac{2\sqrt{2}-\sqrt{6}}{2-\left(\sqrt{3}-1\right)}\)(Vì \(\sqrt{3}>1>0\))
\(=\frac{2\sqrt{2}+\sqrt{6}}{3+\sqrt{3}}+\frac{2\sqrt{2}-\sqrt{6}}{2-\sqrt{3}+1}\)
\(=\frac{2\sqrt{2}+\sqrt{6}}{3+\sqrt{3}}+\frac{2\sqrt{2}-\sqrt{6}}{3-\sqrt{3}}\)
\(=\frac{\left(2\sqrt{2}+\sqrt{6}\right)\left(3-\sqrt{3}\right)}{\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)}+\frac{\left(2\sqrt{2}-\sqrt{6}\right)\left(3+\sqrt{3}\right)}{\left(3-\sqrt{3}\right)\left(3+\sqrt{3}\right)}\)
\(=\frac{6\sqrt{2}-2\sqrt{6}+3\sqrt{6}-3\sqrt{2}}{3^2-\left(\sqrt{3}\right)^2}+\frac{6\sqrt{2}+2\sqrt{6}-3\sqrt{6}-3\sqrt{2}}{3^2-\left(\sqrt{3}\right)^2}\)
\(=\frac{3\sqrt{2}+\sqrt{6}+3\sqrt{2}-\sqrt{6}}{9-3}\)
\(=\frac{6\sqrt{2}}{6}=\sqrt{2}\)
