\[{\text{Вычислите:}}\]
$%\frac{{\sqrt[6]{2} \cdot \sqrt[3]{{2\sqrt 8 }}}}{{\left( {\sqrt[4]{4} - 1} \right) \cdot \left( {\sqrt[4]{4} + 1} \right)}}$%
\[\begin{array}{l}1){\text{ }}\sqrt[3]{{2\sqrt 8 }} = {\left( {2 \cdot {8^{\frac{1}{2}}}} \right)^{\frac{1}{3}}} = {\left( {2 \cdot {{\left( {{2^3}} \right)}^{\frac{1}{2}}}} \right)^{\frac{1}{3}}} = \\{\left( {2 \cdot {2^{\frac{3}{2}}}} \right)^{\frac{1}{3}}} = {\left( {{2^{\frac{5}{2}}}} \right)^{\frac{1}{3}}} = {2^{\frac{5}{2} \cdot \frac{1}{3}}} = {2^{\frac{5}{6}}}\\2){\text{ }}\sqrt[6]{2} \cdot \sqrt[3]{{2\sqrt 8 }} = {2^{\frac{1}{6}}} \cdot {2^{\frac{5}{6}}} = {2^{\frac{1}{6} + \frac{5}{6}}} = {2^1} = 2\\3){\text{ }}\left( {\sqrt[4]{4} - 1} \right) \cdot \left( {\sqrt[4]{4} + 1} \right) = {\left( {{4^{\frac{1}{4}}}} \right)^2} - 1 = \\{4^{\frac{1}{2}}} - 1 = \sqrt 4 - 1 = 2 - 1 = 1\\4){\text{ }}\frac{{\sqrt[6]{2} \cdot \sqrt[3]{{2\sqrt 8 }}}}{{\left( {\sqrt[4]{4} - 1} \right) \cdot \left( {\sqrt[4]{4} + 1} \right)}} = \frac{2}{1} = 2\end{array}\]
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