Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 80

\[\boxed{\mathbf{80.}}\]

\[1)\ a^{\frac{1}{9}} \bullet \sqrt[6]{a^{3}\sqrt[3]{a}} =\]

\[= a^{\frac{1}{9}} \bullet \sqrt[6]{a^{3}} \bullet \sqrt[{6 \bullet 3}]{a} = a^{\frac{1}{9}} \bullet a^{\frac{3}{6}} \bullet a^{\frac{1}{18}} =\]

\[= a^{\frac{1}{9} + \frac{3}{6} + \frac{1}{18}} = a^{\frac{2 + 3 + 1}{18}} =\]

\[= a^{\frac{6}{18}} = a^{\frac{1}{3}}\]

\[2)\ b^{\frac{1}{12}} \bullet \sqrt[3]{b\sqrt[4]{b}} = b^{\frac{1}{12}} \bullet \sqrt[3]{b} \bullet \sqrt[{3 \bullet 4}]{b} =\]

\[= b^{\frac{1}{12}} \bullet b^{\frac{1}{3}} \bullet b^{\frac{1}{12}} = b^{\frac{1}{12} + \frac{1}{3} + \frac{1}{12}} =\]

\[= b^{\frac{1 + 4 + 1}{12}} = b^{\frac{6}{12}} = b^{\frac{1}{2}}\]

\[3)\ \left( \sqrt[3]{ab^{- 2}} + \left( \text{ab} \right)^{- 6} \right) \bullet \sqrt[6]{ab^{4}} =\]

\[= a^{\frac{1}{3} + \frac{1}{6}} \bullet b^{- \frac{2}{3} + \frac{4}{6}} + a^{- \frac{1}{6} + \frac{1}{6}} \bullet b^{- \frac{1}{6} + \frac{4}{6}} =\]

\[= a^{\frac{2 + 1}{6}} \bullet b^{\frac{- 4 + 4}{6}} + a^{0} \bullet b^{\frac{3}{6}} =\]

\[= a^{\frac{3}{6}} \bullet b^{0} + 1 \bullet b^{\frac{1}{2}} = a^{\frac{1}{2}} \bullet 1 + b^{\frac{1}{2}} =\]

\[= a^{\frac{1}{2}} + b^{\frac{1}{2}}\]

\[= \left( a^{\frac{1}{3}} \right)^{3} + \left( b^{\frac{1}{3}}\ \right)^{3} = a + b\]