Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 493

\[\boxed{\mathbf{493}.}\]

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

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

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

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

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

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

\[= \frac{\left( \sqrt{a} - \sqrt{b} \right)^{4}}{|a|};\]

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

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

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

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

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

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

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

\[= \frac{\sqrt[3]{\text{ab}}}{\sqrt[3]{a} + \sqrt[3]{b}};\]

\[3)\ \frac{a^{\frac{1}{4}} - a^{\frac{9}{4}}}{a^{\frac{1}{4}} - a^{\frac{5}{4}}} - \frac{b^{- \frac{1}{2}} - b^{\frac{3}{2}}}{b^{\frac{1}{2}} + b^{- \frac{1}{2}}} =\]

\[= \frac{a^{\frac{1}{4}} \bullet \left( 1 - a^{\frac{8}{4}} \right)}{a^{\frac{1}{4}} \bullet \left( 1 - a^{\frac{4}{4}} \right)} -\]

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

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

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

\[- \frac{(1 - b)(1 + b)}{1 + b} =\]

\[= (1 + a) - (1 - b) =\]

\[= 1 + a - 1 + b = a + b;\]

\[4)\ \frac{\sqrt{a} - a^{- \frac{1}{2}}b}{1 - \sqrt{a^{- 1}b}} - \frac{\sqrt[3]{a^{2}} - a^{- \frac{1}{3}}b}{\sqrt[6]{a} + a^{- \frac{1}{3}}\sqrt{b}} =\]

\[= \frac{a^{\frac{1}{2}} - a^{- \frac{1}{2}}b}{a^{0} - a^{- \frac{1}{2}}b^{\frac{1}{2}}} - \frac{a^{\frac{2}{3}} - a^{- \frac{1}{3}}b}{a^{\frac{1}{6}} + a^{- \frac{1}{3}}b^{\frac{1}{2}}} =\]

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

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

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

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

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

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

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

\[= 2b^{\frac{1}{2}} = 2\sqrt{b};\]