1) \(\sqrt{6+2\sqrt{5}}=\sqrt{5+2\sqrt{5}+1}=\sqrt{\left(\sqrt{5}\right)^2+2\sqrt{5}+1^2}=\sqrt{\left(\sqrt{5}+1\right)^2}=\sqrt{5}+1\)
3) \(\sqrt{15-6\sqrt{6}}-\sqrt{10-4\sqrt{6}}\)
\(=\sqrt{15-2\cdot3\sqrt{6}}-\sqrt{10-2\cdot2\sqrt{6}}\)
\(=\sqrt{9-2\cdot3\sqrt{6}+6}-\sqrt{4-2\cdot2\sqrt{6}+6}\)
\(=\sqrt{3^2-2\cdot3\sqrt{6}+\left(\sqrt{6}\right)^2}-\sqrt{2^2-2\cdot2\sqrt{6}+\left(\sqrt{6}\right)^2}\)
\(=\sqrt{\left(3-\sqrt{6}\right)^2}-\sqrt{\left(2-\sqrt{6}\right)^2}\)
\(=\left|3-\sqrt{6}\right|-\left|2-\sqrt{6}\right|\)
\(=\left(3-\sqrt{6}\right)-\left(\sqrt{6}-2\right)\)
\(=3-\sqrt{6}-\sqrt{6}+2\)
\(=5-2\sqrt{6}\)
5) \(\sqrt{31-10\sqrt{6}}-\sqrt{\left(3-2\sqrt{6}\right)^2}\)
\(=\sqrt{25-2\cdot5\sqrt{6}+6}-\left|3-2\sqrt{6}\right|\)
\(=\sqrt{5^2-2\cdot5\sqrt{6}+\left(\sqrt{6}\right)^2}-\left(2\sqrt{6}-3\right)\)
\(=\sqrt{\left(5-\sqrt{6}\right)^2}-2\sqrt{6}+3\)
\(=\left|5-\sqrt{6}\right|-2\sqrt{6}+3\)
\(=5-\sqrt{6}-2\sqrt{6}+3\)
\(=8-3\sqrt{6}\)