ГДЗ по алгебре и начала математического анализа 10 класс Алимов Задание 41

\[\boxed{\mathbf{41.}}\]

\[1)\ \sqrt[5]{a^{6}b^{7}}\ :\sqrt[5]{ab^{2}} = \sqrt[5]{\frac{a^{6}b^{7}}{ab^{2}}} =\]

\[= \sqrt[5]{a^{5}b^{5}} = ab\]

\[2)\ \sqrt[3]{81x^{4}y}\ :\sqrt[3]{3xy} = \sqrt[3]{\frac{81x^{4}y}{3xy}} =\]

\[= \sqrt[3]{27x^{3}} = \sqrt[3]{3^{3}x^{3}} = 3x\]

\[3)\ \sqrt[3]{\frac{3x}{y^{2}}}\ :\sqrt[3]{\frac{y}{9x^{2}}} = \sqrt[3]{\frac{3x}{y^{2}}\ :\frac{y}{9x^{2}}} =\]

\[= \sqrt[3]{\frac{3x}{y^{2}} \bullet \frac{9x^{2}}{y}} = \sqrt[3]{\frac{27x^{3}}{y^{3}}} =\]

\[= \sqrt[3]{\frac{3^{3}x^{3}}{y^{3}}} = \frac{3x}{y}\]

\[4)\ \sqrt[4]{\frac{2b}{a^{3}}}\ :\sqrt[4]{\frac{a}{8b^{3}}} = \sqrt[4]{\frac{2b}{a^{3}}\ :\frac{a}{8b^{3}}} =\]

\[= \sqrt[4]{\frac{2b}{a^{3}} \bullet \frac{8b^{3}}{a}} = \sqrt[4]{\frac{16b^{4}}{a^{4}}} =\]

\[= \sqrt[4]{\frac{2^{4}b^{4}}{a^{4}}} = \frac{2b}{a}\]