1)

\(M=\frac{6+4\sqrt{2}}{\sqrt{2}+\sqrt{6+4\sqrt{2}}}+\frac{6-4\sqrt{2}}{\sqrt{2}-\sqrt{6-4\sqrt{2}}}\)

\(=\frac{6+4\sqrt{2}}{\sqrt{2}+\sqrt{4+2.2.\sqrt{2}+2}}+\frac{6-4\sqrt{2}}{\sqrt{2}-\sqrt{4-2.2.\sqrt{2}+2}}\)

\(=\frac{6+4\sqrt{2}}{\sqrt{2}+\sqrt{\left(2+\sqrt{2}\right)^2}}+\frac{6-4\sqrt{2}}{\sqrt{2}-\sqrt{\left(2-\sqrt{2}\right)^2}}\)

\(=\frac{6+4\sqrt{2}}{2+2\sqrt{2}}+\frac{6-4\sqrt{2}}{-2+2\sqrt{2}}\)

\(=\frac{2.\left(3+2\sqrt{2}\right)}{2.\left(1+\sqrt{2}\right)}+\frac{2.\left(3-2\sqrt{2}\right)}{2.\left(\sqrt{2}-1\right)}\)

\(=\frac{3+2\sqrt{2}}{\sqrt{2}+1}+\frac{3-2\sqrt{2}}{\sqrt{2}-1}\)

\(=\frac{\left(3+2\sqrt{2}\right)\left(\sqrt{2}-1\right)}{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}+\frac{\left(3-2\sqrt{2}\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}\)

\(=1+\sqrt{2}+\sqrt{2}-1=2\sqrt{2}\)

a/ \(\sqrt{6+2\sqrt{2}\sqrt{3-\left(\sqrt{3}+1\right)^2}}=\sqrt{6+2\sqrt{2}\sqrt{2-\sqrt{3}}}\)

\(=\sqrt{6+2\sqrt{4-2\sqrt{3}}}=\sqrt{6+2\sqrt{\left(\sqrt{3}-1\right)^2}}\)

\(=\sqrt{6+2\left(\sqrt{3}-1\right)}=\sqrt{4+2\sqrt{3}}=\sqrt{3}+1\)

b/ \(=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)\sqrt{8-2\sqrt{15}}\)

\(=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}\)

\(=\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)^2=\left(4+\sqrt{15}\right)\left(8-2\sqrt{15}\right)\)

\(=2\left(4+\sqrt{15}\right)\left(4+\sqrt{15}\right)=2\left(16-15\right)\)

\(M=\sqrt{\frac{\left(3\sqrt{3}-4\right)\left(2\sqrt{3}-1\right)}{\left(2\sqrt{3}+1\right)\left(2\sqrt{3}-1\right)}}+\sqrt{\frac{\left(\sqrt{3}+4\right)\left(5+2\sqrt{3}\right)}{\left(5+2\sqrt{3}\right)\left(5-2\sqrt{3}\right)}}\)

\(M=\sqrt{\frac{18-3\sqrt{3}-8\sqrt{3}+4}{11}}+\sqrt{\frac{5\sqrt{3}+6+20+8\sqrt{3}}{13}}\)

\(M=\sqrt{\frac{11\left(2-\sqrt{3}\right)}{11}}+\sqrt{\frac{13\left(2+\sqrt{3}\right)}{13}}=\sqrt{2-\sqrt{3}}+\sqrt{2+\sqrt{3}}\)

\(M=\frac{1}{\sqrt{2}}\left(\sqrt{4-2\sqrt{3}}+\sqrt{4+2\sqrt{3}}\right)\)

\(M=\frac{1}{\sqrt{2}}\left(\sqrt{\left(\sqrt{3}-1\right)^2}+\sqrt{\left(\sqrt{3}+1\right)^2}\right)\)

\(M=\frac{1}{\sqrt{2}}\left(\sqrt{3}-1+\sqrt{3}+1\right)=\frac{2\sqrt{3}}{\sqrt{2}}=\sqrt{6}\)

a) Ta có: \(\left(\frac{3}{2}\sqrt{6}+2\sqrt{\frac{2}{3}}-4\sqrt{\frac{3}{2}}\right)\left(3\sqrt{\frac{2}{3}}-\sqrt{12}-\sqrt{6}\right)\)

\(=\left(\sqrt{\frac{9}{4}\cdot6}+\sqrt{4\cdot\frac{2}{3}}-\sqrt{16\cdot\frac{3}{2}}\right)\left(\sqrt{9\cdot\frac{2}{3}}-2\sqrt{3}-\sqrt{6}\right)\)

\(=\left(\sqrt{\frac{27}{2}}+\sqrt{2}-2\sqrt{6}\right)\cdot\left(\sqrt{6}-2\sqrt{3}-\sqrt{6}\right)\)

\(=-2\sqrt{3}\cdot\left(\sqrt{\frac{27}{2}}+\sqrt{2}-2\sqrt{6}\right)\)

\(=-\sqrt{12\cdot\frac{27}{2}}-2\sqrt{6}+4\sqrt{18}\)

\(=-9\sqrt{2}-2\sqrt{6}+12\sqrt{2}\)

\(=3\sqrt{2}-2\sqrt{6}\)

b) Ta có: \(\frac{4}{\sqrt{3}+1}-\frac{5}{\sqrt{3}-2}+\frac{6}{\sqrt{3}-3}\)

\(=\frac{4\left(\sqrt{3}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}-\frac{5\left(\sqrt{3}+2\right)}{\left(\sqrt{3}-2\right)\left(\sqrt{3}+2\right)}+\frac{6\left(\sqrt{3}+3\right)}{\left(\sqrt{3}-3\right)\left(\sqrt{3}+3\right)}\)

\(=\frac{4\left(\sqrt{3}-1\right)}{2}-\frac{5\left(\sqrt{3}+2\right)}{-1}+\frac{6\left(\sqrt{3}+3\right)}{-6}\)

\(=2\left(\sqrt{3}-1\right)+5\left(\sqrt{3}+2\right)-\left(\sqrt{3}+3\right)\)

\(=2\sqrt{3}-2+5\sqrt{3}+10-\sqrt{3}-3\)

\(=6\sqrt{3}+5\)