Страница 33 номер 81, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение
Упростить выражение (80–83). 1) \(\left(1 - 2\sqrt{\frac{b}{a}} + \frac{b}{a}\right) : \left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)^2\); 2) \(\left(a^{\frac{1}{3}} + b^{\frac{1}{3}}\right) : \left(2 + \sqrt[3]{\frac{a}{b}} + \sqrt[3]{\frac{b}{a}}\right)\); 3) \(\frac{a^{\frac{1}{4}} - a^{\frac{9}{4}}}{a^{\frac{1}{4}} - a^{\frac{5}{4}}} - \frac{b^{-\frac{1}{2}} - b^{\frac{3}{2}}}{b^{\frac{1}{2}} + b^{-\frac{1}{2}}}\); 4) \(\frac{\sqrt{a} - a^{-\frac{1}{2}}b}{1 - \sqrt{a^{-1}b}} - \frac{\sqrt[3]{a^2} - a^{-\frac{1}{3}}b}{\sqrt[6]{a} + a^{-\frac{1}{3}}\sqrt{b}}\).
1) \(\left(1 - 2\sqrt{\frac{b}{a}} + \frac{b}{a}\right) : \left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)^{2} = \left(1 - \left(\frac{b}{a}\right)^{\frac{1}{2}}\right)^{2} : \left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)^{2} =\)
\(= \left(\dfrac{a^{\frac{1}{2}} - b^{\frac{1}{2}}}{a^{\frac{1}{2}}}\right)^{2} : \left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)^{2} = \dfrac{1}{a};\)
2) \(\left(a^{\frac{1}{3}} + b^{\frac{1}{3}}\right) : \left(2 + \sqrt[3]{\frac{a}{b}} + \sqrt[3]{\frac{b}{a}}\right) = \left(a^{\frac{1}{3}} + b^{\frac{1}{3}}\right) : \left(\left(\frac{a}{b}\right)^{\frac{1}{6}} + \left(\frac{b}{a}\right)^{\frac{1}{6}}\right)^{2} =\)
\(= \left(a^{\frac{1}{3}} + b^{\frac{1}{3}}\right) : \left(\dfrac{a^{\frac{1}{3}} + b^{\frac{1}{3}}}{(ab)^{\frac{1}{6}}}\right)^{2} = \dfrac{(ab)^{\frac{1}{3}}}{a^{\frac{1}{3}} + b^{\frac{1}{3}}} = \dfrac{\sqrt[3]{ab}}{\sqrt[3]{a} + \sqrt[3]{b}};\)
3) \(\frac{a^{\frac{1}{4}} - a^{\frac{9}{4}}}{a^{\frac{1}{4}} - a^{\frac{5}{4}}} - \frac{b^{-\frac{1}{2}} - b^{\frac{3}{2}}}{b^{\frac{1}{2}} + b^{-\frac{1}{2}}} = \frac{a^{\frac{1}{4}}(1 - a^{2})}{a^{\frac{1}{4}}(1 - a)} - \frac{b^{-\frac{1}{2}}(1 - b^{2})}{b^{-\frac{1}{2}}(b + 1)} = (1 + a) - (1 - b) = a + b;\)
4) \(\frac{\sqrt{a} - a^{-\frac{1}{2}}b}{1 - \sqrt{a^{-1}b}} - \frac{\sqrt[3]{a^{2}} - a^{-\frac{1}{3}}b}{\sqrt[6]{a} + a^{-\frac{1}{3}}\sqrt{b}} = \frac{a^{\frac{1}{2}} - a^{-\frac{1}{2}}b}{1 - a^{-\frac{1}{2}}b^{\frac{1}{2}}} - \frac{a^{\frac{2}{3}} - a^{-\frac{1}{3}}b}{a^{\frac{1}{6}} + a^{-\frac{1}{3}}b^{\frac{1}{2}}} =\)
\(= \frac{a^{-\frac{1}{2}}(a - b)}{a^{-\frac{1}{2}}\left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)} - \frac{a^{-\frac{1}{3}}(a - b)}{a^{-\frac{1}{3}}\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right)} =\)
\(= \dfrac{\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right)(a - b) - \left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)(a - b)}{a - b} =\)
\(= a^{\frac{1}{2}} + b^{\frac{1}{2}} - a^{\frac{1}{2}} + b^{\frac{1}{2}} = 2b^{\frac{1}{2}} = 2\sqrt{b}.\)
