\(\frac{3\sqrt{2}+2\sqrt{3}}{\sqrt{2}+\sqrt{3}}-\frac{5}{1+\sqrt{6}}\)
\(=\frac{\sqrt{6}\left(\sqrt{2}+\sqrt{3}\right)}{\sqrt{2}+\sqrt{3}}-\frac{5\left(\sqrt{6}-1\right)}{\left(\sqrt{6}-1\right)\left(\sqrt{6}+1\right)}\)
\(=\sqrt{6}-\frac{5\left(\sqrt{6}-1\right)}{5}\)
\(=\sqrt{6}-\sqrt{6}+1\)
\(=1\)
\(\frac{1}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{1}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)
\(=\frac{\sqrt{2}}{2+\sqrt{4+2\sqrt{3}}}+\frac{\sqrt{2}}{2-\sqrt{4-2\sqrt{3}}}\)
\(=\frac{\sqrt{2}}{2+\sqrt{\left(1+\sqrt{3}\right)^2}}+\frac{\sqrt{2}}{2-\sqrt{\left(1-\sqrt{3}\right)^2}}\)
\(=\sqrt{2}\left(\frac{1}{3+\sqrt{3}}+\frac{1}{3-\sqrt{3}}\right)\)
\(=\sqrt{2}\left(\frac{3+\sqrt{3}+3-\sqrt{3}}{\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)}\right)\)
\(=\sqrt{2}.\frac{6}{6}\)
\(=\sqrt{2}\)
