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[tex]a)\ \ 3\sqrt{6}\cdot2\sqrt{8}=3\cdot2\cdot\sqrt{6\cdot8}=6\sqrt{48}=6\sqrt{16\cdot3}=6\cdot4\sqrt{3}=24\sqrt{3}\\\\\\b)\ \ 4\sqrt{15}\cdot2\sqrt{5}=4\cdot2\cdot\sqrt{15\cdot5}=8\sqrt{75}=8\sqrt{25\cdot3}=8\cdot5\sqrt{3}=40\sqrt{3}\\\\\\c)\ \ \frac{1}{2}\sqrt{8}\cdot5\sqrt{6}=\frac{1}{2}\cdot5\cdot\sqrt{8\cdot6}=\frac{5}{2}\sqrt{48}=\frac{5}{2}\sqrt{16\cdot3}=\frac{5}{\not2_{1}}\cdot\not4^2\sqrt{3}=5\cdot2\sqrt{3}=10\sqrt{3}[/tex]
[tex]d)\ \ 5\sqrt{3}+3\sqrt{12}=5\sqrt{3}+3\sqrt{4\cdot3}=5\sqrt{3}+3\cdot2\sqrt{3}=5\sqrt{3}+6\sqrt{3}=11\sqrt{3}\\\\\\e)\ \ 5\sqrt{27}+7\sqrt{12}=5\sqrt{9\cdot3}+7\sqrt{4\cdot3}=5\cdot3\sqrt{3}+7\cdot2\sqrt{3}=15\sqrt{3}+14\sqrt{3}=29\sqrt{3}\\\\\\f)\ \ 2\sqrt{48}+\frac{1}{3}\sqrt{27}=2\sqrt{16\cdot3}+\frac{1}{3}\sqrt{9\cdot3}=2\cdot4\sqrt{3}+\frac{1}{\not3}\cdot\not3\sqrt{3}=8\sqrt{3}+\sqrt{3}=9\sqrt{3}[/tex]