Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 490

\[\boxed{\mathbf{490}.}\]

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

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

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

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

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

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

\[3)\ \frac{a^{\frac{5}{3}}b^{- \frac{4}{3}} - a^{\frac{1}{3}}}{a^{\frac{5}{3}}b^{\frac{1}{3}} - a^{\frac{1}{3}}b^{\frac{5}{3}}} =\]

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

\[= b^{- \frac{4}{3} - \frac{1}{3}} = b^{- \frac{5}{3}};\]

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

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

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