$\displaystyle \begin{array}{{>{\displaystyle}l}} 1.\ 5\sqrt{12} -4\sqrt{3} +\sqrt{48} =10\sqrt{3} -4\sqrt{3} +4\sqrt{3} =10\sqrt{3}\\ 2.\ \frac{2}{3}\sqrt{3} -\frac{3}{4}\sqrt{2} +\frac{\sqrt{2}}{5} -\frac{\sqrt{3}}{2} =\frac{\sqrt{3}}{3} -\frac{11}{20}\sqrt{2}\\ 3.\ \frac{\sqrt{5} +\sqrt{2}}{5-2} +\frac{\sqrt{5} -\sqrt{2}}{5-2} =\frac{2\sqrt{5}}{3}\\ 4.\ \left( 2\sqrt{6} -4\sqrt{3} +5\sqrt{2} -\sqrt{3}\right) .3\sqrt{6}\\ =\left( 2\sqrt{6} +5\sqrt{2} -5\sqrt{3}\right) 3\sqrt{6}\\ =36+30\sqrt{3} -45\sqrt{2}\\ 5.\ \left[\frac{15\left(\sqrt{6} -1\right)}{6-1} +\frac{4\left(\sqrt{6} +2\right)}{6-4} -\frac{12\left( 3+\sqrt{6}\right)}{9-6}\right]\left(\sqrt{6} +11\right)\\ =\left(\sqrt{6} -11\right)\left(\sqrt{6} +11\right) =6-11^{2} =-115\\ 6.\ \sqrt{\left( 2-\sqrt{5}\right)^{2}} -\sqrt{5} =\sqrt{5} -2-\sqrt{5} =-2\\ 7.\ \sqrt{\left(\sqrt{2} -1\right)^{2}} -\sqrt{\left( 2+\sqrt{2}\right)^{2}} =\sqrt{2} -1-2-\sqrt{2} =-3\\ 8.\ \frac{-4\left(\sqrt{3} -1\right)}{3-1} -\frac{1+\sqrt{2}}{1-2} +\frac{10\left( 2\sqrt{2} +3\right)}{8-3}\\ =\frac{-4\sqrt{3} +4}{2} +\frac{1+\sqrt{2}}{1} +\frac{20\sqrt{2} +30}{5}\\ =\frac{-20\sqrt{3} +20+10+10\sqrt{2} +40\sqrt{2} +60}{10}\\ =\frac{-20\sqrt{3} +50\sqrt{2} +90}{10} =-2\sqrt{3} +3\sqrt{2} +9\\ 9.\ \frac{\sqrt{xy}\left(\sqrt{x} +\sqrt{y}\right)}{\left(\sqrt{x} +\sqrt{y}\right)\left(\sqrt{x} -\sqrt{y}\right)} -\frac{x+\sqrt{xy} +y}{\left(\sqrt{x} -\sqrt{y}\right)\left( x+\sqrt{xy} +y\right)}\\ =\frac{\sqrt{xy}}{\sqrt{x} -\sqrt{y}} -\frac{1}{\sqrt{x} -\sqrt{y}} =\frac{\sqrt{xy} -1}{\sqrt{x} -\sqrt{y}}\\ 10.\ \sqrt{2}\sqrt{\frac{3}{2} -\frac{\sqrt{5}}{2}} -\ \sqrt{2}\sqrt{\frac{3}{2} +\frac{\sqrt{5}}{2}}\\ =\ \sqrt{2}\left(\sqrt{\frac{5}{4} -2.\frac{1}{2} .\frac{\sqrt{5}}{2} +\frac{1}{4}} -\sqrt{\frac{5}{4} +2.\frac{1}{2} .\frac{\sqrt{5}}{2} +\frac{1}{4}}\right)\\ =\sqrt{2}\left(\sqrt{\left(\frac{\sqrt{5}}{2} -\frac{1}{2}\right)^{2}} -\sqrt{\left(\frac{\sqrt{5}}{2} +\frac{1}{2}\right)^{2}}\right)\\ =\sqrt{2}\left(\frac{\sqrt{5}}{2} -\frac{1}{2} -\frac{\sqrt{5}}{2} -\frac{1}{2}\right) =-\sqrt{2} \end{array}$