Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 478

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

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

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

\[= a^{\frac{1}{4}} + b^{\frac{1}{4}};\]

\[2)\ \frac{m^{\frac{1}{2}} + n^{\frac{1}{2}}}{m + 2\sqrt{\text{mn}} + n} =\]

\[= \frac{m^{\frac{1}{2}} + n^{\frac{1}{2}}}{\left( m^{\frac{1}{2}} \right)^{2} + 2m^{\frac{1}{2}}n^{\frac{1}{2}} + \left( n^{\frac{1}{2}} \right)^{2}} =\]

\[= \frac{m^{\frac{1}{2}} + n^{\frac{1}{2}}}{\left( m^{\frac{1}{2}} + n^{\frac{1}{2}} \right)^{2}\ } = \frac{1}{m^{\frac{1}{2}} + n^{\frac{1}{2}}} =\]

\[= \left( m^{\frac{1}{2}} + n^{\frac{1}{2}} \right)^{- 1};\]

\[3)\ \frac{c - 2c^{\frac{1}{2}} + 1}{\sqrt{c} - 1} =\]

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

\[= \frac{\left( c^{\frac{1}{2}} - 1 \right)^{2}\ }{c^{\frac{1}{2}} - 1} = c^{\frac{1}{2}} - 1 = \sqrt{c} - 1;\ \]

\[4)\ \frac{x^{\frac{2}{3}} - 2x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}}}{x - y} =\]

\[= \frac{\left( x^{\frac{1}{3}} - y^{\frac{1}{3}} \right)^{2}}{\left( x^{\frac{1}{3}} - y^{\frac{1}{3}} \right)\left( x^{\frac{2}{3}} + x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}} \right)} =\]

\[= \frac{x^{\frac{1}{3}} - y^{\frac{1}{3}}}{x^{\frac{2}{3}} + x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}}}.\]