$\displaystyle \begin{array}{{>{\displaystyle}l}} M=\sqrt{4-\sqrt{7}} -\sqrt{4+\sqrt{7}} +\sqrt{2} \ \\ M=\frac{\sqrt{8-2\sqrt{7}} -\sqrt{8+2\sqrt{7}} +2}{\sqrt{2}}\\ M=\frac{\sqrt{\left( 1-2\sqrt{7} +7\right)} -\sqrt{1+2\sqrt{7} +7} +2}{\sqrt{2}}\\ M=\frac{\sqrt{\left( 1-\sqrt{7}\right)^{2}} -\sqrt{\left( 1+\sqrt{7}\right)^{2}} +2}{\sqrt{2}}\\ M=\frac{\sqrt{7} -1-1-\sqrt{7} +2}{\sqrt{2}} =\frac{0}{\sqrt{2}} =0\ \\ b) 4\sqrt{3+2\sqrt{2}} -\sqrt{56\sqrt{2} +81}\\ =4\sqrt{1+2\sqrt{2} +2} -\sqrt{32+2.4\sqrt{2} .7+49} \ \\ =4\sqrt{\left( 1+\sqrt{2}\right)^{2}} -\sqrt{\left( 4\sqrt{2} +7\right)^{2}}\\ =4\left( 1+\sqrt{2}\right) -\left( 4\sqrt{2} +7\right)\\ =4+4\sqrt{2} -4\sqrt{2} -7\\ =-3\\ c) \ P=\sqrt{3-\sqrt{5}}\left( 3+\sqrt{5}\right)\left(\sqrt{10} -\sqrt{2}\right)\\ P=\sqrt{3-\sqrt{5} .}\sqrt{3+\sqrt{5}} .\sqrt{3+\sqrt{5}} .\sqrt{2}\left(\sqrt{5} -1\right)\\ P=\sqrt{\left( 3-\sqrt{5}\right)\left( 3+\sqrt{5}\right)} .\sqrt{6+2\sqrt{5}}\left(\sqrt{5} -1\right)\\ P=\sqrt{9-5} .\sqrt{1+2\sqrt{5} +5}\left(\sqrt{5} -1\right)\\ P=2.\sqrt{\left( 1+\sqrt{5}\right)^{2}} .\left(\sqrt{5} -1\right)\\ P=2.\left( 1+\sqrt{5}\right)\left(\sqrt{5} -1\right)\\ P=2.( 5-1) =8\\ Q=\sqrt{10+\sqrt{24} +\sqrt{40} +\sqrt{60} \ }\\ Q=\sqrt{2+3+5+2\sqrt{6} +2\sqrt{10} +2\sqrt{15}}\\ Q=\sqrt{2+3+5+2\left(\sqrt{2.3} +\sqrt{2.5} +\sqrt{3.5}\right)}\\ Q=\sqrt{\left(\sqrt{2} +\sqrt{3} +\sqrt{5}\right)^{2}}\\ Q=\sqrt{2} +\sqrt{3} +\sqrt{5} \end{array}$
