Simplify. \[ \left(-2 a b^{4}\right)^{4} \] Write your answer without parentheses.

To simplify \(\left(-2 a b^{4}\right)^{4}\), we apply the power of a product rule, which states that \((xy)^n = x^n y^n\). First, we break it down: \[ \left(-2\right)^{4} \cdot \left(a\right)^{4} \cdot \left(b^{4}\right)^{4} \] Calculating each part: \(-2\) raised to the 4th power is: \[ (-2)^{4} = 16 \] Next, \(a\) raised to the 4th power is: \[ (a)^{4} = a^{4} \] Then, \(b^{4}\) raised to the 4th power is: \[ (b^{4})^{4} = b^{16} \] Now combine all parts together: \[ \left(-2 a b^{4}\right)^{4} = 16 \cdot a^{4} \cdot b^{16} \] Thus, the final answer is: \[ \boxed{16 a^{4} b^{16}} \]