Страница 34 номер 87, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение

1) \(\frac{a^{\frac{3}{2}}}{\sqrt{a} + \sqrt{b}} - \frac{ab^{\frac{1}{2}}}{\sqrt{b} - \sqrt{a}} - \frac{2a^{2}}{a - b} =\)
\(= \frac{a^{\frac{3}{2}}\left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right) + ab^{\frac{1}{2}}\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right) - 2a^{2}}{a - b} = \frac{a^{2} - a^{\frac{3}{2}}b^{\frac{1}{2}} + a^{\frac{3}{2}}b^{\frac{1}{2}} + ab - 2a^{2}}{a - b}\)
\(= \dfrac{ab - a^{2}}{a - b} = -\dfrac{a(a - b)}{a - b} = -a;\)

2) \(\frac{3xy - y^{2}}{x - y} - \frac{y\sqrt{y}}{\sqrt{x} - \sqrt{y}} - \frac{y\sqrt{x}}{\sqrt{x} + \sqrt{y}} = \frac{3xy - y^{2} - y^{\frac{3}{2}}\left(x^{\frac{1}{2}} + y^{\frac{1}{2}}\right) - yx^{\frac{1}{2}}\left(x^{\frac{1}{2}} - y^{\frac{1}{2}}\right)}{x - y} =\)
\(= \dfrac{3xy - y^{2} - y^{\frac{3}{2}}x^{\frac{1}{2}} - y^{2} - yx + y^{\frac{3}{2}}x^{\frac{1}{2}}}{x - y} =\)
\(= \dfrac{2xy - 2y^{2}}{x - y} = \dfrac{2y(x - y)}{x - y} = 2y;\)

3) \(\frac{1}{\sqrt[3]{a} + \sqrt[3]{b}} - \frac{\sqrt[3]{a} + \sqrt[3]{b}}{a^{\frac{2}{3}} - \sqrt[3]{ab} + b^{\frac{2}{3}}} = \frac{a^{\frac{2}{3}} - \sqrt[3]{ab} + b^{\frac{2}{3}} - \left(\sqrt[3]{a} + \sqrt[3]{b}\right)^{2}}{\left(\sqrt[3]{a} + \sqrt[3]{b}\right)\left(a^{\frac{2}{3}} - \sqrt[3]{ab} + b^{\frac{2}{3}}\right)} =\)
\(= \frac{a^{\frac{2}{3}} - \sqrt[3]{ab} + b^{\frac{2}{3}} - a^{\frac{2}{3}} - 2\sqrt[3]{ab} - b^{\frac{2}{3}}}{a + b} =\)
\(= -\dfrac{3\sqrt[3]{ab}}{a + b};\)

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