\(1,\sqrt{6+2\sqrt{5}}=\sqrt{\sqrt{5^2}+2\sqrt{5}+1}=\sqrt{\left(\sqrt{5}+1\right)^2}=\left|\sqrt{5}+1\right|=\sqrt{5}+1\\ 3,\sqrt{15-6\sqrt{6}}-\sqrt{10-4\sqrt{6}}\\ =\sqrt{\sqrt{6^2}-2.3\sqrt{6}+3^2}-\sqrt{\sqrt{6^2}-2.2\sqrt{6}+2^2}\\ =\sqrt{\left(\sqrt{6}-3\right)^2}-\sqrt{ \left(\sqrt{6}-2\right)^2}\\ =\left|\sqrt{6}-3\right|-\left|\sqrt{6}-2\right|\\ =3-\sqrt{6}-\sqrt{6}+2\\ =-2\sqrt{6}+5\)
\(5,\sqrt{31-10\sqrt{6}}-\sqrt{\left(3-2\sqrt{6}\right)^2}\\ =\sqrt{\sqrt{6^2}-2.5\sqrt{6}+5^2}-\left|3-2\sqrt{6}\right|\\ =\sqrt{\left(\sqrt{6}-5\right)^2}-\left(2\sqrt{6}-3\right)\\ =\left|\sqrt{6}-5\right|-2\sqrt{6}+3\\ =5-\sqrt{6}-2\sqrt{6}+3\\ =8-3\sqrt{6}\)
