5.
[tex]a) \ 2^{\frac{2}{3}}\cdot2^{\frac{4}{3}}=2^{\frac{2}{3}+\frac{4}{3}} = 2^{\frac{6}{3}} = 2^{2} = 4\\\\b) \ 5^{\frac{4}{3}}\cdot5^{\frac{5}{3}} = 5^{\frac{4}{3}+\frac{5}{3}} = 5^{\frac{9}{3}} = 5^{3} = 125\\\\c) \ 4^{\frac{9}{2}}:4^{\frac{5}{2}}=4^{\frac{9}{2}-\frac{5}{2}} = 4^{\frac{4}{2}} = 4^{2} = 16\\\\d) \ 2^{7,5}:2^{2,5} = 2^{7,7-2,5} = 2^{5} = 32\\\\e) \ 3^{\frac{1}{2}}\cdot3^{-\frac{1}{3}}\cdot3^{\frac{1}{6}} = 3^{\frac{1}{2}+(-\frac{1}{3})+\frac{1}{6}}=3^{\frac{1}{3}} = \sqrt[3]{3}[/tex]
6.
[tex]a) \ 12^{\frac{3}{2}}\cdot3^{\frac{3}{2}}=(12\cdot3)^{\frac{3}{2}} = 36^{\frac{3}{2}} = (6^{2})^{\frac{3}{2}} = 6^{3} = 216[/tex]
[tex]b) \ 12^{\frac{5}{2}}:3^{\frac{5}{2}} = (12:3)^{\frac{5}{2}} = 4^{\frac{5}{2}} = (2^{2})^{\frac{5}{2}}=2^{5} = 32[/tex]
[tex]c) \ 24^{\frac{2}{3}}\cdot2^{\frac{1}{3}}=\sqrt[3]{24^{2}}\cdot\sqrt[3]{3}=\sqrt[3]{(8\cdot3)^{2}\cdot3}=\sqrt[3]{64\cdot3^{3}} = \sqrt[3]{4^{3}\cdot3^{3}} = 4\cdot3 = 12[/tex]
[tex]d) \ 9^{\frac{4}{3}}:24^{\frac{2}{3}} = \sqrt[3]{9^{4}}:\sqrt[3]{24^{2}} = \sqrt[3]{81^{2}:24^{2}} = \sqrt[3]{(\frac{81}{24})^{2}} = \sqrt[3]{(\frac{27}{8})^{2}} =\frac{3}{2}\cdot\frac{3}{2} = \frac{9}{4} = 2\frac{1}{4}[/tex]
[tex]e) \ 375^{\frac{1}{3}}\cdot3^{\frac{2}{3}} = \sqrt[3]{375}\cdot\sqrt[3]{3^{2}} = \sqrt[3]{375\cdot9}=\sqrt[3]{125\cdot3\cdot9} = \sqrt[3]{125\cdot27} = 5\cdot3 = 15[/tex]
