We can solve this problem by using the properties of exponents and radicals. First, we can rewrite the equation \(sqrt[3]{4^{2x}}=16\) as \((4^{2x})^{1/3}=16\). Then, we can use the property that \((a^b)^c=a^{bc}\) to simplify the left side of the equation:

\((4^{2x})^{1/3}=4^{(2x)(1/3)}=4^{(2/3)x}\)

Now, we can rewrite the right side of the equation as a power of 4: \(16=4^2\). So, the equation becomes:

\(4^{(2/3)x}=4^2\)

Since the bases on both sides of the equation are equal, we can equate their exponents:

\((2/3)x=2\)

Solving this equation for x, we get x = 3.