5.
[tex]a) \ 9^{\frac{5}{6}}\cdot9^{\frac{2}{3}} =9^{\frac{5}{6}+\frac{2}{3}} = 9^{\frac{5}{6}+\frac{4}{6}} = 9^{\frac{9}{6}} = 9^{\frac{3}{2}}=(3^{2})^{\frac{3}{2}} = 3^{3} = 27[/tex]
[tex]b) \ 8^{\frac{4}{5}}:8^\frac{2}{15}}=8^{\frac{12}{15}}:8^{\frac{2}{15}} = 8^{\frac{12-2}{15}} = 8^{\frac{10}{15}} = 8^{\frac{2}{3}} = (2^{3})^{\frac{2}{3}} = 2^{2} = 4[/tex]
[tex]c) \ (\frac{2}{5})^{-\frac{1}{2}}\cdot(\frac{5}{18})^{-\frac{1}{2}}=(\frac{5}{2})^{\frac{1}{2}}\cdot(\frac{18}{5})^{\frac{1}{2}} = (\frac{5}{2}\cdot\frac{18}{5})^{\frac{1}{2}} = 9^{\frac{1}{2}} = (3^{2})^{\frac{1}{2}} = 3[/tex]
[tex]d) \ (2\frac{2}{3})^{1\frac{1}{2}}:(\frac{2}{3})^{1\frac{1}{2}} = (\frac{8}{3}\cdot\frac{3}{2})^{\frac{3}{2}} = 4^{\frac{3}{2}} = (2^{2})^{\frac{3}{2}} = 2^{3} = 8[/tex]
[tex]e) \ ((\frac{4}{9})^{\frac{3}{4}})^{-\frac{2}{3}} = (\frac{4}{9})^{\frac{3}{4}\cdot(-\frac{2}{3})}=(\frac{4}{9})^{-\frac{1}{2}} = (\frac{9}{4})^{\frac{1}{2}} = ((\frac{3}{2})^{2})^\frac{1}{2}}=\frac{3}{2} = 1\frac{1}{2}[/tex]
