1: \(\sqrt{\left(3-\sqrt5\right)^2}+\sqrt{\left(2-\sqrt5\right)^2}\)
\(=3-\sqrt5+\left|2-\sqrt5\right|\)
\(=3-\sqrt5+\sqrt5-2=3-2=1\)
2: \(\sqrt{\left(\sqrt5-\sqrt6\right)^2}+\sqrt{11+2\sqrt{30}}\)
\(=\left|\sqrt5-\sqrt6\right|+\sqrt{\left(\sqrt5+\sqrt6\right)^2}\)
\(=\sqrt6-\sqrt5+\sqrt5+\sqrt6=2\sqrt6\)
3: \(\sqrt{\left(2\sqrt2-3\right)^2}+\sqrt{11-6\sqrt2}\)
\(=3-2\sqrt2+\sqrt{11-2\cdot3\sqrt2}\)
\(=3-2\sqrt2+\sqrt{\left(3-\sqrt2\right)^2}=3-2\sqrt2+3-\sqrt2=6-3\sqrt2\)
4: \(\sqrt{\left(2-\sqrt3\right)^2}+\sqrt{4-2\sqrt3}\)
\(=\left|2-\sqrt3\right|+\sqrt{\left(\sqrt3-1\right)^2}\)
\(=2-\sqrt3+\sqrt3-1=2-1=1\)
5: \(\sqrt{\left(6-2\sqrt5\right)^2}-\sqrt{101-36\cdot\sqrt5}\)
\(=\left|6-2\sqrt5\right|-\sqrt{\left(9-2\sqrt5\right)^2}\)
\(=6-2\sqrt5-9+2\sqrt5=6-9=-3\)
6: \(\sqrt{\left(4-3\sqrt2\right)^2}-\sqrt{19+6\sqrt2}\)
\(=\left|4-3\sqrt2\right|-\sqrt{\left(3\sqrt2+1\right)^2}\)
\(=4-3\sqrt2-3\sqrt2+1=5-6\sqrt2\)
7: \(\sqrt{\left(3-5\sqrt2\right)^2}-\sqrt{51+10\sqrt2}\)
\(=\left|3-5\sqrt2\right|-\sqrt{51+2\cdot5\sqrt2}\)
\(=5\sqrt2-3-\sqrt{\left(5\sqrt2+1\right)^2}=5\sqrt2-3-5\sqrt2-1=-3-1=-4\)
8: \(\sqrt{3+2\sqrt2}-\sqrt{3-2\sqrt2}\)
\(=\sqrt{\left(\sqrt2+1\right)^2}-\sqrt{\left(\sqrt2-1\right)^2}\)
\(=\sqrt2+1-\sqrt2+1=2\)
9: \(\sqrt{15-6\sqrt6}+\sqrt{33-12\sqrt6}\)
\(=\sqrt{\left(3-\sqrt6\right)^2}+\sqrt{\left(2\sqrt6-3\right)^2}\)
\(=3-\sqrt6+2\sqrt6-3=\sqrt6\)
