Answer
$\frac{2}{3}$
Work Step by Step
Re-write $\sqrt[3]{16}=16^{\frac{1}{3}}$.
$x=\log_4{16}^{\frac{1}{3}}$
Write expression in exponential form.
$4^x=16^{\frac{1}{3}}$
Re-write both sides in terms of the same base.
$4^x=(4^2)^{\frac{1}{3}}$
Set exponents equal. (If $x^a=x^b$, than $a=b$.)
$x=2(\frac{1}{3})=\frac{2}{3}$
