ГДЗ по алгебре и начала математического анализа 10 класс Колягин Задание 540

\[\boxed{\mathbf{540}.}\]

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

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

\[- \sqrt{a} =\]

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

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

\[= \sqrt{a} - 1 - \sqrt{a} = - 1.\]

\[2)\ \frac{\sqrt[3]{a} - a^{\frac{7}{3}}}{a^{\frac{1}{3}} - \sqrt[3]{a^{4}}} + \frac{a^{- \frac{1}{3}} - \left( \sqrt[3]{a} \right)^{5}}{a^{\frac{2}{3}} - \left( \sqrt[3]{a} \right)^{- 1}} =\]

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

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

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

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

\[= 1 + a - 1 - a = 0.\]