Решебник по алгебре и начала математического анализа 10 класс Алимов Задание 114

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\[1)\ \frac{\sqrt{x} - \sqrt{y}}{\sqrt[4]{x} - \sqrt[4]{y}} - \frac{\sqrt{x} + \sqrt[4]{\text{xy}}}{\sqrt[4]{x} + \sqrt[4]{y}} =\]

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

\[= x^{\frac{1}{4}} + y^{\frac{1}{4}} - x^{\frac{1}{4}} = y^{\frac{1}{4}} = \sqrt[4]{y}\]

\[2)\ \frac{x - y}{\sqrt[3]{x} - \sqrt[3]{y}} - \frac{x + y}{\sqrt[3]{x} + \sqrt[3]{y}} =\]

\[= 2x^{\frac{1}{3}}y^{\frac{1}{3}} = 2\sqrt[3]{\text{xy}}\]

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

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

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

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

\[4)\ \frac{x\sqrt{x} - y\sqrt{y}}{x\sqrt{y} - y\sqrt{x}} - 1 =\]

\[= \frac{x^{1} \bullet x^{\frac{1}{2}} - y^{1} \bullet y^{\frac{1}{2}}}{x^{1} \bullet y^{\frac{1}{2}} - y^{1} \bullet x^{\frac{1}{2}}} - 1 =\]

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

\[= \frac{x + \sqrt{\text{xy}} + y}{\sqrt{\text{xy}}} - 1 =\]

\[= \frac{x + \sqrt{\text{xy}} + y - \sqrt{\text{xy}}}{\sqrt{\text{xy}}} = \frac{x + y}{\sqrt{\text{xy}}}\]