Solution by e_power_pi_times_i
$\sqrt[3]{4}$ and $\sqrt[4]{8}$ can be expressed as $2^{\frac{2}{3}}$ and $2^{\frac{3}{4}}$, so their product is $2^{\frac{17}{12}} = \boxed{\textbf{(E) } 2\sqrt[12]{32}}$.
The product of $\sqrt[3]{4}$ and $\sqrt[4]{8}$ equals
$\textbf{(A) }\sqrt[7]{12}\qquad \textbf{(B) }2\sqrt[7]{12}\qquad \textbf{(C) }\sqrt[7]{32}\qquad \textbf{(D) }\sqrt[12]{32}\qquad \textbf{(E) }2\sqrt[12]{32}$