5. Найдите значение выражения $$\left(\frac{2^{\frac{1}{3}} \cdot 24^{\frac{1}{2}}}{\sqrt[12]{2}}\right)^2$$

Преобразуем выражение:

$$\left(\frac{2^{\frac{1}{3}} \cdot 24^{\frac{1}{2}}}{\sqrt[12]{2}}\right)^2 = \left(\frac{2^{\frac{1}{3}} \cdot (2^3 \cdot 3)^{\frac{1}{2}}}{2^{\frac{1}{12}}}\right)^2 = \left(\frac{2^{\frac{1}{3}} \cdot 2^{\frac{3}{2}} \cdot 3^{\frac{1}{2}}}{2^{\frac{1}{12}}}\right)^2 =$$ $$=\left(\frac{2^{\frac{1}{3} + \frac{3}{2}} \cdot 3^{\frac{1}{2}}}{2^{\frac{1}{12}}}\right)^2 = \left(\frac{2^{\frac{2+9}{6}} \cdot 3^{\frac{1}{2}}}{2^{\frac{1}{12}}}\right)^2 = \left(\frac{2^{\frac{11}{6}} \cdot 3^{\frac{1}{2}}}{2^{\frac{1}{12}}}\right)^2 = \left(2^{\frac{11}{6} - \frac{1}{12}} \cdot 3^{\frac{1}{2}}\right)^2 =$$ $$=\left(2^{\frac{22-1}{12}} \cdot 3^{\frac{1}{2}}\right)^2 = \left(2^{\frac{21}{12}} \cdot 3^{\frac{1}{2}}\right)^2 = \left(2^{\frac{7}{4}} \cdot 3^{\frac{1}{2}}\right)^2 = 2^{\frac{7}{4} \cdot 2} \cdot 3^{\frac{1}{2} \cdot 2} = 2^{\frac{7}{2}} \cdot 3 = 2^{3,5} \cdot 3 = 2^3 \cdot 2^{0,5} \cdot 3 =$$ $$= 8 \cdot \sqrt{2} \cdot 3 = 24\sqrt{2}$$

Ответ: $$24\sqrt{2}$$