Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 457

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\[1)\ \sqrt[3]{\sqrt[3]{a^{18}}} + \left( \sqrt{\sqrt[3]{a^{4}}} \right)^{3} =\]

\[= \sqrt[{3 \bullet 3}]{a^{18}} + \sqrt[{2 \bullet 3}]{a^{4 \bullet 3}} = \sqrt[9]{a^{18}} +\]

\[+ \sqrt[6]{a^{12}} = \sqrt[9]{a^{9 \bullet 2}} + \sqrt[6]{a^{6 \bullet 2}} =\]

\[= a^{2} + a^{2} = 2a^{2};\]

\[2)\ \left( \sqrt{\sqrt[3]{x^{2}}} \right)^{3} + 2\left( \sqrt[4]{\sqrt{x}} \right)^{8} =\]

\[= \sqrt[{2 \bullet 3}]{x^{2 \bullet 3}} + 2\sqrt[{4 \bullet 2}]{x^{8}} =\]

\[= \sqrt[6]{x^{6}} + 2\sqrt[8]{x^{8}} = x + 2x = 3x;\]

\[3)\ \sqrt[3]{\sqrt{x^{6}y^{12}}} - \left( \sqrt[5]{xy^{2}} \right)^{5} =\]

\[= \sqrt[{3 \bullet 2}]{x^{6}y^{12}} - \sqrt[5]{\left( xy^{2} \right)^{5}} =\]

\[= \sqrt[6]{x^{6} \bullet y^{6 \bullet 2}} - xy^{2} =\]

\[= xy^{2} - xy^{2} = 0;\]

\[4)\ \left( \left( \sqrt[5]{a\sqrt[5]{a}} \right)^{5} - \sqrt[5]{a} \right)\ :\sqrt[10]{a^{2}} =\]

\[= \left( \left( \sqrt[5]{a} \right)^{5} \bullet \left( \sqrt[5]{\sqrt[5]{a}} \right)^{5} - \sqrt[5]{a} \right)\ :\]

\[:\sqrt[{2 \bullet 5}]{a^{2}} =\]

\[= \left( \sqrt[5]{a^{5}} \bullet \sqrt[{5 \bullet 5}]{a^{5}} - \sqrt[5]{a} \right)\ :\sqrt[5]{a} =\]

\[= \left( a \bullet \sqrt[5]{a} - \sqrt[5]{a} \right)\ :\sqrt[5]{a} =\]

\[= a \bullet \sqrt[5]{\frac{a}{a}} - \sqrt[5]{\frac{a}{a}} =\]

\[= a \bullet \sqrt[5]{1} - \sqrt[5]{1} =\]

\[= a \bullet 1 - 1 = a - 1.\]