Sửa đề: \(\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)
Ta có: \(\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)
\(=\frac{\left(2+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{2+\sqrt{3}}\right)}{\left(\sqrt{2}+\sqrt{2+\sqrt{3}}\right)\left(\sqrt{2}-\sqrt{2+\sqrt{3}}\right)}+\frac{\left(2-\sqrt{3}\right)\left(\sqrt{2}+\sqrt{2-\sqrt{3}}\right)}{\left(\sqrt{2}-\sqrt{2-\sqrt{3}}\right)\left(\sqrt{2}+\sqrt{2-\sqrt{3}}\right)}\)
\(=\frac{2\sqrt{2}-\sqrt{4+2\sqrt{3}}+\sqrt{6}-\sqrt{6+3\sqrt{3}}}{2-\left(2+\sqrt{3}\right)}+\frac{2\sqrt{2}+\sqrt{4-2\sqrt{3}}-\sqrt{6}-\sqrt{6-3\sqrt{3}}}{2-\left(2-\sqrt{3}\right)}\)
\(=\frac{2\sqrt{2}-\sqrt{\left(\sqrt{3}+1\right)^2}+\sqrt{6}-\sqrt{6+3\sqrt{3}}}{-\sqrt{3}}+\frac{2\sqrt{2}+\sqrt{\left(\sqrt{3}-1\right)^2}-\sqrt{6}-\sqrt{6-3\sqrt{3}}}{\sqrt{3}}\)
\(=\frac{-2\sqrt{2}+\left|\sqrt{3}+1\right|-\sqrt{6}+\sqrt{6+3\sqrt{3}}+2\sqrt{2}+\left|\sqrt{3}-1\right|-\sqrt{6}-\sqrt{6-3\sqrt{3}}}{\sqrt{3}}\)
\(=\frac{\sqrt{3}+1-2\sqrt{6}+\sqrt{3}-1+\sqrt{6+3\sqrt{3}}-\sqrt{6-3\sqrt{3}}}{\sqrt{3}}\)
\(=\frac{2\sqrt{6}-4\sqrt{3}+\sqrt{12+6\sqrt{3}}-\sqrt{12-6\sqrt{3}}}{\sqrt{6}}\)
\(=\frac{2\sqrt{6}-4\sqrt{3}+\sqrt{\left(3+\sqrt{3}\right)^2}-\sqrt{\left(3-\sqrt{3}\right)^2}}{\sqrt{6}}\)
\(=\frac{2\sqrt{6}-4\sqrt{3}+3+\sqrt{3}-3+\sqrt{3}}{\sqrt{6}}\)
\(=\frac{2\sqrt{6}-2\sqrt{3}}{\sqrt{6}}\)
\(=2-\sqrt{2}\)