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МатематикаРешебник по алгебре и начала математического анализа 10 класс Колягин Задание 538
Задание 538
\[\boxed{\mathbf{538}.}\]
\[1)\ \frac{\sqrt{x} - \sqrt{y}}{\sqrt[4]{x} - \sqrt[4]{y}} - \frac{\sqrt{x} + \sqrt[4]{\text{xy}}}{\sqrt[4]{x} + \sqrt[4]{y}} =\]
\[= \frac{x^{\frac{1}{2}} - y^{\frac{1}{2}}}{x^{\frac{1}{4}} - y^{\frac{1}{4}}} - \frac{x^{\frac{1}{2}} + x^{\frac{1}{4}}y^{\frac{1}{4}}}{x^{\frac{1}{4}} + y^{\frac{1}{4}}} =\]
\[= \frac{\left( x^{\frac{1}{4}} - y^{\frac{1}{4}} \right)\left( x^{\frac{1}{4}} + y^{\frac{1}{4}} \right)}{x^{\frac{1}{4}} - y^{\frac{1}{4}}} -\]
\[- \frac{x^{\frac{1}{4}}\left( x^{\frac{1}{4}} + y^{\frac{1}{4}} \right)}{x^{\frac{1}{4}} + y^{\frac{1}{4}}} = x^{\frac{1}{4}} + y^{\frac{1}{4}} -\]
\[- x^{\frac{1}{4}} = y^{\frac{1}{4}} = \sqrt[4]{y};\]
\[2)\ \frac{x - y}{\sqrt[3]{x} - \sqrt[3]{y}} - \frac{x + y}{\sqrt[3]{x} + \sqrt[3]{y}} =\]
\[= \frac{(x - y)\left( x^{\frac{2}{3}} + x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}} \right)}{\left( x^{\frac{1}{3}} - y^{\frac{1}{3}} \right)\left( x^{\frac{2}{3}} + x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}} \right)} -\]
\[- \frac{(x + y)\left( x^{\frac{2}{3}} - x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}} \right)}{\left( x^{\frac{1}{3}} + y^{\frac{1}{3}} \right)\left( x^{\frac{2}{3}} - x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}} \right)} =\]
\[= \frac{(x - y)\left( x^{\frac{2}{3}} + x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}} \right)}{x - y} -\]
\[- \frac{(x + y)\left( x^{\frac{2}{3}} - x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}} \right)}{x + y} =\]
\[= x^{\frac{2}{3}} + x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}} -\]
\[- \left( x^{\frac{2}{3}} - x^{\frac{1}{3}}y^{\frac{1}{3}} + y^{\frac{2}{3}} \right) =\]
\[= 2x^{\frac{1}{3}}y^{\frac{1}{3}} = 2\sqrt[3]{\text{xy}};\]
\[3)\ \frac{\sqrt{x} - \sqrt[3]{y^{2}}}{\sqrt[4]{x} + \sqrt[3]{y}} + \sqrt[3]{y} = \frac{x^{\frac{1}{2}} - y^{\frac{2}{3}}}{x^{\frac{1}{4}} + y^{\frac{1}{3}}} +\]
\[+ y^{\frac{1}{3}} = \frac{\left( x^{\frac{1}{4}} - y^{\frac{1}{3}} \right)\left( x^{\frac{1}{4}} + y^{\frac{1}{3}} \right)}{x^{\frac{1}{4}} + y^{\frac{1}{3}}} +\]
\[+ y^{\frac{1}{3}} =\]
\[= x^{\frac{1}{4}} - y^{\frac{1}{3}} + y^{\frac{1}{3}} = x^{\frac{1}{4}} = \sqrt[4]{x};\]
\[4)\ \frac{x\sqrt{x} - y\sqrt{y}}{x\sqrt{y} - y\sqrt{x}} - 1 =\]
\[= \frac{x^{1} \bullet x^{\frac{1}{2}} - y^{1} \bullet y^{\frac{1}{2}}}{x^{1} \bullet y^{\frac{1}{2}} - y^{1} \bullet x^{\frac{1}{2}}} - 1 =\]
\[= \frac{x^{\frac{3}{2}} - y^{\frac{3}{2}}}{x^{\frac{1}{2}}y^{\frac{1}{2}} \bullet \left( x^{\frac{1}{2}} - y^{\frac{1}{2}} \right)} - 1 =\]
\[= \frac{\left( x^{\frac{1}{2}} - y^{\frac{1}{2}} \right)\left( x^{\frac{2}{2}} + x^{\frac{1}{2}}x^{\frac{1}{2}} + y^{\frac{2}{2}} \right)}{x^{\frac{1}{2}}y^{\frac{1}{2}} \bullet \left( x^{\frac{1}{2}} - y^{\frac{1}{2}} \right)} -\]
\[- 1 = \frac{x + \sqrt{\text{xy}} + y}{\sqrt{\text{xy}}} - 1 =\]
\[= \frac{x + \sqrt{\text{xy}} + y - \sqrt{\text{xy}}}{\sqrt{\text{xy}}} = \frac{x + y}{\sqrt{\text{xy}}}.\]
