Страница 38 номер 117, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение

1) \(\left(\frac{\left(\sqrt[4]{a} + \sqrt[4]{b}\right)^{2} + \left(\sqrt[4]{a} - \sqrt[4]{b}\right)^{2}}{a + \sqrt{ab}}\right)^{5} \cdot \sqrt[3]{a^{10} \cdot \sqrt{a}} =\)
\(= \left(\dfrac{\sqrt{a} + 2\sqrt[4]{ab} + \sqrt{b} + \sqrt{a} - 2\sqrt[4]{ab} + \sqrt{b}}{a + \sqrt{ab}}\right)^{5} \cdot \sqrt[3]{\sqrt{a^{21}}} =\)
\(= \left(\dfrac{2\sqrt{a} + 2\sqrt{b}}{\sqrt{a}\left(\sqrt{a} + \sqrt{b}\right)}\right)^{5}a^{\frac{21}{6}} = \left(\dfrac{2}{\sqrt{a}}\right)^{5}a^{\frac{7}{2}} =\)
\(= 32a^{\frac{7}{2} - \frac{5}{2}} = 32a;\)

2) \(\left(\frac{a - a^{-1}}{\left(\sqrt[3]{a^{-1}} + \sqrt[3]{a} + 1\right)\left(\sqrt[3]{a^{-1}} - \sqrt[3]{a} + 1\right)} + \sqrt[3]{a^{-1}}\right)^{-3} =\)
\(= \left(\dfrac{a - a^{-1}}{a^{-\frac{2}{3}} - a^{0} + a^{-\frac{1}{3}} + a^{0} - a^{\frac{2}{3}} + a^{\frac{1}{3}} + a^{-\frac{1}{3}} - a^{\frac{1}{3}} + 1} + a^{-\frac{1}{3}}\right)^{-3} =\)
\(= \left(\dfrac{a - a^{-1} + a^{-\frac{1}{3}}\left(a^{-\frac{2}{3}} + 2a^{-\frac{1}{3}} - a^{\frac{2}{3}} + 1\right)}{a^{-\frac{2}{3}} + 2a^{-\frac{1}{3}} - a^{\frac{2}{3}} + 1}\right)^{-3} =\)
\(= \left(\dfrac{a - a^{-1} + a^{-1} + 2a^{-\frac{2}{3}} - a^{\frac{1}{3}} + a^{-\frac{1}{3}}}{a^{-\frac{2}{3}} + 2a^{-\frac{1}{3}} - a^{\frac{2}{3}} + 1}\right)^{-3} =\)
\(= \left(\dfrac{a + 2a^{-\frac{2}{3}} - a^{\frac{1}{3}} + a^{-\frac{1}{3}}}{a^{-\frac{2}{3}} + 2a^{-\frac{1}{3}} - a^{\frac{2}{3}} + 1}\right)^{-3} =\)
\(= \left(\dfrac{a^{-\frac{1}{3}}\left(a^{\frac{4}{3}} + 2a^{-\frac{1}{3}} - a^{\frac{2}{3}} + 1\right)}{a^{-\frac{2}{3}} + 2a^{-\frac{1}{3}} - a^{\frac{2}{3}} + 1}\right)^{-3} =\)
\(= \left(\dfrac{a^{\frac{5}{3}} + 2 - a + a^{\frac{1}{3}}}{1 + 2a^{\frac{1}{3}} - a^{\frac{4}{3}} + a^{\frac{2}{3}}}\right)^{-3} =\)
\(= \left(\dfrac{\left(a^{\frac{2}{3}} + a^{\frac{1}{3}} + 1\right)\left(a - a^{\frac{2}{3}} - a^{\frac{1}{3}} + 2\right)}{\left(1 + a^{\frac{1}{3}} + a^{\frac{2}{3}}\right)\left(1 + a^{\frac{1}{3}} - a^{\frac{2}{3}}\right)}\right)^{-3} =\)
\(= \left(\dfrac{a - a^{\frac{2}{3}} - a^{\frac{1}{3}} + 2}{1 + a^{\frac{1}{3}} - a^{\frac{2}{3}}}\right)^{-3} = \left(\dfrac{1 + a^{\frac{1}{3}} - a^{\frac{2}{3}}}{a - a^{\frac{2}{3}} - a^{\frac{1}{3}} + 2}\right)^{3};\)

3) \(\left(\frac{a^{\frac{3}{2}} - b^{\frac{3}{2}}}{\sqrt{a} - \sqrt{b}} - \sqrt{\frac{ab\sqrt[3]{a} + ab^{\frac{4}{3}}}{a^{\frac{1}{3}} + b^{\frac{1}{3}}}}\right) \cdot \frac{1}{a + b} =\)
\(= \left(\frac{\left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)\left(a + (ab)^{\frac{1}{2}} + b\right)}{a^{\frac{1}{2}} - b^{\frac{1}{2}}} - \sqrt{\frac{ab\left(a^{\frac{1}{3}} + b^{\frac{1}{3}}\right)}{a^{\frac{1}{3}} + b^{\frac{1}{3}}}}\right) \cdot \frac{1}{a + b} =\)
\(= \left(a + \sqrt{ab} + b - \sqrt{ab}\right) \cdot \dfrac{1}{a + b} = 1.\)