Страница 34 номер 88, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение
Упростить выражение (87–89). 1) \(\frac{a - b}{\sqrt[3]{a} - \sqrt[3]{b}} - \frac{a + b}{a^{\frac{1}{3}} + b^{\frac{1}{3}}}\); 2) \(\frac{a + b}{a^{\frac{2}{3}} - a^{\frac{1}{3}}b^{\frac{1}{3}} + b^{\frac{2}{3}}} - \frac{a - b}{a^{\frac{2}{3}} + a^{\frac{1}{3}}b^{\frac{1}{3}} + b^{\frac{2}{3}}}\); 3) \(\frac{a^{\frac{2}{3}} + b^{\frac{2}{3}}}{a - b} - \frac{1}{a^{\frac{1}{3}} - b^{\frac{1}{3}}}\); 4) \(\frac{a^{\frac{1}{3}} - b^{\frac{1}{3}}}{a + b} - \frac{1}{a^{\frac{2}{3}} - a^{\frac{1}{3}}b^{\frac{1}{3}} + b^{\frac{2}{3}}}\).
1) \(\frac{a - b}{\sqrt[3]{a} - \sqrt[3]{b}} - \frac{a + b}{a^{\frac{1}{3}} + b^{\frac{1}{3}}} = \frac{(a - b)\left(a^{\frac{1}{3}} + b^{\frac{1}{3}}\right) - (a + b)\left(a^{\frac{1}{3}} - b^{\frac{1}{3}}\right)}{a^{\frac{2}{3}} - b^{\frac{2}{3}}} =\)
\(= \dfrac{a^{\frac{4}{3}} + ab^{\frac{1}{3}} - a^{\frac{1}{3}}b - b^{\frac{4}{3}} - a^{\frac{4}{3}} + ab^{\frac{1}{3}} - a^{\frac{1}{3}}b + b^{\frac{4}{3}}}{a^{\frac{2}{3}} - b^{\frac{2}{3}}} =\)
\(= \dfrac{2ab^{\frac{1}{3}} - 2a^{\frac{1}{3}}b}{a^{\frac{2}{3}} - b^{\frac{2}{3}}} = \dfrac{2a^{\frac{1}{3}}b^{\frac{1}{3}}\left(a^{\frac{2}{3}} - b^{\frac{2}{3}}\right)}{a^{\frac{2}{3}} - b^{\frac{2}{3}}} = 2a^{\frac{1}{3}}b^{\frac{1}{3}};\)
2) \(\frac{a + b}{a^{\frac{2}{3}} - a^{\frac{1}{3}}b^{\frac{1}{3}} + b^{\frac{2}{3}}} - \frac{a - b}{a^{\frac{2}{3}} + a^{\frac{1}{3}}b^{\frac{1}{3}} + b^{\frac{2}{3}}} =\)
\(= \frac{(a + b)\left(a^{\frac{1}{3}} + b^{\frac{1}{3}}\right)}{\left(a^{\frac{1}{3}} + b^{\frac{1}{3}}\right)\left(a^{\frac{2}{3}} - a^{\frac{1}{3}}b^{\frac{1}{3}} + b^{\frac{2}{3}}\right)} - \frac{(a - b)\left(a^{\frac{1}{3}} - b^{\frac{1}{3}}\right)}{\left(a^{\frac{1}{3}} - b^{\frac{1}{3}}\right)\left(a^{\frac{2}{3}} + a^{\frac{1}{3}}b^{\frac{1}{3}} + b^{\frac{2}{3}}\right)} =\)
\(= \dfrac{(a + b)\left(a^{\frac{1}{3}} + b^{\frac{1}{3}}\right)}{a + b} - \dfrac{(a - b)\left(a^{\frac{1}{3}} - b^{\frac{1}{3}}\right)}{a - b} =\)
\(= a^{\frac{1}{3}} + b^{\frac{1}{3}} - a^{\frac{1}{3}} + b^{\frac{1}{3}} = 2\sqrt[3]{b};\)
3) \(\frac{a^{\frac{2}{3}} + b^{\frac{2}{3}}}{a - b} - \frac{1}{a^{\frac{1}{3}} - b^{\frac{1}{3}}} = \frac{a^{\frac{2}{3}} + b^{\frac{2}{3}}}{a - b} - \frac{a^{\frac{2}{3}} + a^{\frac{1}{3}}b^{\frac{1}{3}} + b^{\frac{2}{3}}}{a - b} = -\frac{a^{\frac{1}{3}}b^{\frac{1}{3}}}{a - b};\)
4) \(\frac{a^{\frac{1}{3}} - b^{\frac{1}{3}}}{a + b} - \frac{1}{a^{\frac{2}{3}} - a^{\frac{1}{3}}b^{\frac{1}{3}} + b^{\frac{2}{3}}} = \frac{a^{\frac{1}{3}} - b^{\frac{1}{3}}}{a + b} - \frac{a^{\frac{1}{3}} + b^{\frac{1}{3}}}{a + b} = -\frac{2\sqrt[3]{b}}{a + b}.\)
