Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 539

\[\boxed{\mathbf{539}.}\]

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

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

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

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

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

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

\[2)\ \left( \frac{3x^{\frac{2}{3}} + 5x^{\frac{1}{3}}}{x + 1} + \frac{1}{x^{\frac{1}{3}} + 1} \right)\ :\]

\[:\left( 4x^{\frac{1}{3}} + 4 + \frac{1}{x^{\frac{1}{3}}} \right) =\]

\[= \frac{3x^{\frac{2}{3}} + 5x^{\frac{1}{3}}{+ x}^{\frac{2}{3}} - x^{\frac{1}{3}} + 1}{x + 1}\ \ :\]

\[:\frac{4x^{\frac{2}{3}} + 4x^{\frac{1}{3}} + 1}{x^{\frac{1}{3}}} =\]

\[= \frac{4x^{\frac{2}{3}} + 4x^{\frac{1}{3}} + 1}{x + 1} \cdot\]

\[\cdot \frac{x^{\frac{1}{3}}}{4x^{\frac{2}{3}} + 4x^{\frac{1}{3}} + 1} = \frac{x^{\frac{1}{3}}}{x + 1}.\]