Решебник по алгебре и начала математического анализа 10 класс Алимов Задание 88

\[\boxed{\mathbf{88}\mathbf{.}}\]

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

\[= \frac{2ab^{\frac{1}{3}} - 2ba^{\frac{1}{3}}}{a^{\frac{2}{3}} - b^{\frac{2}{3}}} =\]

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

\[= 2\sqrt[3]{\text{ab}}\]

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

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

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

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

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

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

\[= - \frac{2\sqrt[3]{b}}{a + b}\]