Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 115

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\[1)\ \ \left( \frac{a^{\frac{4}{3}}b + ab^{\frac{4}{3}}}{a^{\frac{1}{3}} + b^{\frac{1}{3}}} \bullet \frac{1}{a^{\frac{1}{3}}b^{\frac{1}{3}}} \right)^{3} =\]

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

\[= \left( \frac{\text{ab}}{a^{\frac{1}{3}}b^{\frac{1}{3}}} \right)^{3} = \left( a^{1 - \frac{1}{3}} \bullet b^{1 - \frac{1}{3}} \right)^{3} =\]

\[= \left( a^{\frac{2}{3}} \bullet b^{\frac{2}{3}} \right)^{3} = a^{2}b^{2}\]

\[2)\ \frac{a^{\frac{1}{3}} - b^{\frac{1}{3}}}{\sqrt[3]{\text{ab}}} \bullet \frac{ab^{\frac{1}{3}} - a^{\frac{1}{3}}b}{\sqrt[3]{a} + \sqrt[3]{b}} =\]

\[= \frac{a^{\frac{1}{3}} - b^{\frac{1}{3}}}{a^{\frac{1}{3}}b^{\frac{1}{3}}} \bullet \frac{a^{\frac{1}{3}}b^{\frac{1}{3}} \bullet \left( a^{\frac{2}{3}} - b^{\frac{2}{3}} \right)}{a^{\frac{1}{3}} + b^{\frac{1}{3}}} =\]

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

\[3)\ \frac{a^{\frac{2}{3}} - b^{\frac{2}{3}}}{a^{\frac{1}{3}} + b^{\frac{1}{3}}} \bullet \frac{a^{\frac{2}{3}} + \sqrt[3]{\text{ab}} + b^{\frac{2}{3}}}{a - b} =\]

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

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

\[= \frac{\left( a^{\frac{2}{3}} - b^{\frac{2}{3}} \right) \bullet \left( \left( a^{\frac{2}{3}} \right)^{3} + \left( b^{\frac{2}{3}} \right)^{3} \right)}{a^{\frac{2}{3}} - b^{\frac{2}{3}}} =\]

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