Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 78

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\[1)\ \frac{a^{\frac{4}{3}} \bullet \left( a^{- \frac{1}{3}} + a^{\frac{2}{3}} \right)}{a^{\frac{1}{4}} \bullet \left( a^{\frac{3}{4}} + a^{- \frac{1}{4}} \right)} =\]

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

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

\[2)\ \frac{b^{\frac{1}{5}} \bullet \left( \sqrt[5]{b^{4}} - \sqrt[5]{b^{- 1}\ } \right)}{b^{\frac{2}{3}} \bullet \left( \sqrt[3]{b} - \sqrt[3]{b^{- 2}} \right)} =\]

\[= \frac{b^{\frac{1}{5}} \bullet \left( b^{\frac{4}{5}} - b^{- \frac{1}{5}} \right)}{b^{\frac{2}{3}} \bullet \left( b^{\frac{1}{3}} - b^{- \frac{2}{3}} \right)} =\]

\[= \frac{b^{\frac{1}{5} + \frac{4}{5}} - b^{\frac{1}{5} - \frac{1}{5}}}{b^{\frac{2}{3} + \frac{1}{3}} - b^{\frac{2}{3} - \frac{2}{3}}} = \frac{b^{1} - b^{0}}{b^{1} - b^{0}} = 1\]

\[3)\ \frac{a^{\frac{5}{3}}b^{- 1} - a^{- \frac{1}{3}}}{\sqrt[3]{a^{2}}} =\]

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

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

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

\[4)\ \frac{a^{\frac{1}{3}}\sqrt{b} + b^{\frac{1}{3}}\sqrt{a}}{\sqrt[6]{a} + \sqrt[6]{b}} =\]

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

\[= \frac{a^{\frac{2}{6}}b^{\frac{2}{6}} \bullet \left( b^{\frac{1}{6}} + a^{\frac{1}{6}} \right)}{a^{\frac{1}{6}} + b^{\frac{1}{6}}} = a^{\frac{1}{3}}b^{\frac{1}{3}}\]