Упрощение алгебраических выражений. Задача 46

ОТВЕТ: \({a^{\frac{2}{3}}}\).

\(\dfrac{{{a^{\frac{4}{3}}}-8{a^{\frac{1}{3}}}b}}{{{a^{\frac{2}{3}}} + 2\sqrt[3]{{ab}} + 4{b^{\frac{2}{3}}}}}{\left( {1-2\sqrt[3]{{\dfrac{b}{a}}}} \right)^{-1}} = \dfrac{{{a^{\frac{1}{3}}}\left( {a-8b} \right)}}{{{{\left( {\sqrt[3]{a}} \right)}^2} + 2\sqrt[3]{a}\sqrt[3]{b} + {{\left( {2\sqrt[3]{b}} \right)}^2}}} \cdot {\left( {1-\dfrac{{2\sqrt[3]{b}}}{{\sqrt[3]{a}}}} \right)^{-1}} = \)

\( = \dfrac{{\sqrt[3]{a}\left( {{{\left( {\sqrt[3]{a}} \right)}^3}-{{\left( {2\sqrt[3]{b}} \right)}^3}} \right)}}{{{{\left( {\sqrt[3]{a}} \right)}^2} + \sqrt[3]{a} \cdot 2\sqrt[3]{b} + {{\left( {2\sqrt[3]{b}} \right)}^2}}} \cdot {\left( {\dfrac{{\sqrt[3]{a}-2\sqrt[3]{b}}}{{\sqrt[3]{a}}}} \right)^{-1}} = \)

\( = \dfrac{{\sqrt[3]{a}\left( {\sqrt[3]{a}-2\sqrt[3]{b}} \right)\left( {{{\left( {\sqrt[3]{a}} \right)}^2} + \sqrt[3]{a} \cdot 2\sqrt[3]{b} + {{\left( {2\sqrt[3]{b}} \right)}^2}} \right)}}{{{{\left( {\sqrt[3]{a}} \right)}^2} + \sqrt[3]{a} \cdot 2\sqrt[3]{b} + {{\left( {2\sqrt[3]{b}} \right)}^2}}} \cdot \dfrac{{\sqrt[3]{a}}}{{\sqrt[3]{a}-2\sqrt[3]{b}}} = \)

\( = \sqrt[3]{a} \cdot \sqrt[3]{a} = {\left( {\sqrt[3]{a}} \right)^2} = {a^{\frac{2}{3}}}.\)

Ответ:  \({a^{\frac{2}{3}}}.\)