Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 86

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\[1)\ \sqrt[3]{10}\text{\ \ }и\ \ \sqrt[5]{20}\]

\[\sqrt[3]{10} = \sqrt[{3 \bullet 5}]{10^{5}} = \sqrt[15]{100\ 000};\]

\[\sqrt[5]{20} = \sqrt[{5 \bullet 3}]{20^{3}} = \sqrt[15]{8000};\]

\[100\ 000 > 8000;\]

\[\sqrt[15]{100\ 000} > \sqrt[15]{8000};\]

\[\sqrt[3]{10} > \sqrt[5]{20}.\]

\[2)\ \sqrt[4]{5}\text{\ \ }и\ \ \sqrt[3]{7};\]

\[\sqrt[4]{5} = \sqrt[{4 \bullet 3}]{5^{3}} = \sqrt[12]{125};\]

\[\sqrt[3]{7} = \sqrt[{3 \bullet 4}]{7^{4}} = \sqrt[12]{2401};\]

\[125 < 2401;;\]

\[\sqrt[12]{125} < \sqrt[12]{2401};\]

\[\ \sqrt[4]{5} < \sqrt[3]{7}.\]

\[3)\ \sqrt{17}\text{\ \ }и\ \ \sqrt[3]{28};\]

\[\sqrt{17} = \sqrt[{2 \bullet 3}]{17^{3}} = \sqrt[6]{4913};\]

\[\sqrt[3]{28} = \sqrt[{3 \bullet 2}]{28^{2}} = \sqrt[6]{784};\]

\[4913 > 784;\]

\[\sqrt[6]{4913} > \sqrt[6]{784};\]

\[\sqrt{17} > \sqrt[3]{28}.\]

\[4)\ \sqrt[3]{13}\text{\ \ }и\ \ \sqrt[5]{23};\ \]

\[\sqrt[3]{13} = \sqrt[{3 \bullet 5}]{13^{5}} = \sqrt[15]{371\ 293};\]

\[\sqrt[5]{23} = \sqrt[{5 \bullet 3}]{23^{3}} = \sqrt[15]{12\ 167};\]

\[371\ 293 > 12\ 167;\]

\[\sqrt[15]{371\ 293} > \sqrt[15]{12\ 167};\]

\[\sqrt[3]{13} > \sqrt[5]{23}.\]