Simplify.

Rewrite the expression in the form
\[4^n\].
\[\left(4^2\right)^{4}=\]

Using the property $(a^m)^n=a^{mn},$ we can simplify as follows: \[\left(4^2\right)^{4}=4^{2\cdot4}=4^8.\]Thus, $\left(4^2\right)^{4}$ is equivalent to $\boxed{4^8}$.