1: \(\frac{5\sqrt6-6\sqrt5}{2\sqrt{15}-5\sqrt2}=\frac{\sqrt{150}-\sqrt{180}}{\sqrt{60}-\sqrt{50}}=\frac{\sqrt{30}\left(\sqrt5-\sqrt6\right)}{\sqrt{10}\left(\sqrt6-\sqrt5\right)}=-\sqrt{\frac{30}{10}}=-\sqrt3\)
2: \(\frac{14-6\sqrt5}{\sqrt5-3}\)
\(=\frac{\left(3-\sqrt5\right)^2}{\sqrt5-3}=\frac{\left(\sqrt5-3\right)^2}{\sqrt5-3}=\sqrt5-3\)
3: \(\frac{9-4\sqrt5}{\sqrt5-2}=\frac{\left(\sqrt5-2\right)^2}{\sqrt5-2}=\sqrt5-2\)
5: \(\frac{\left(\sqrt5+2\right)^2-8\sqrt5}{5\sqrt5-8}=\frac{9+4\sqrt5-8\sqrt5}{5\sqrt5-8}=\frac{9-4\sqrt5}{5\sqrt5-8}\)
\(=\frac{\left(9-4\sqrt5\right)\left(5\sqrt5+8\right)}{125-64}=\frac{45\sqrt5+72-100-32\sqrt5}{61}=\frac{13\sqrt5-28}{61}\)
6: \(\frac{8\sqrt6-\left(\sqrt3+2\sqrt2\right)^2}{4\sqrt2-2\sqrt3}\)
\(=\frac{8\sqrt6-3-8-4\sqrt6}{4\sqrt2-2\sqrt3}=\frac{-11+4\sqrt6}{2\left(2\sqrt2-\sqrt3\right)}=\frac{-\left(2\sqrt2-\sqrt3\right)^2}{2\left(2\sqrt2-\sqrt3\right)}\)
\(=\frac{-\left(2\sqrt2-\sqrt3\right)}{2}\)
7: \(\frac{7}{3\sqrt2-5}-\frac{\sqrt{18}-3\sqrt6}{\sqrt3-1}\)
\(=\frac{7\left(3\sqrt2+5\right)}{\left(3\sqrt2-5\right)\left(3\sqrt2+5\right)}-\frac{3\left(\sqrt2-\sqrt6\right)}{\sqrt3-1}\)
\(=\frac{7\left(3\sqrt2+5\right)}{18-25}+\frac{3\sqrt2\left(\sqrt3-1\right)}{\sqrt3-1}=-\left(3\sqrt2+5\right)+3\sqrt2=-5\)
9: \(\frac{\sqrt{12}-6}{\sqrt8-\sqrt{24}}-\frac{3+\sqrt3}{\sqrt3}-\frac{4}{\sqrt7-1}\)
\(=\frac{\sqrt{12}\left(1-\sqrt3\right)}{\sqrt8\left(1-\sqrt3\right)}-\sqrt3-1-\frac{4\left(\sqrt7+1\right)}{\left(\sqrt7-1\right)\left(\sqrt7+1\right)}\)
\(=\sqrt{\frac32}-\sqrt3-1-\frac{4\left(\sqrt7+1\right)}{7-1}=\frac{\sqrt6}{2}-\sqrt3-1-\frac23\left(\sqrt7+1\right)\)
\(=\frac12\sqrt6-\sqrt3-1-\frac23\sqrt7-\frac23=\frac12\sqrt6-\frac23\sqrt7-\sqrt3-\frac53\)
10: \(\frac{x+y-2\sqrt{xy}}{x\cdot\sqrt{y}-y\cdot\sqrt{x}}=\frac{\left(\sqrt{x}-\sqrt{y}\right)^2}{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}=\frac{\sqrt{x}-\sqrt{y}}{\sqrt{xy}}\)
