Simplify. Express your answer using a single exponent. \[ \left(4 n^{5}\right)^{3} \]

To simplify the expression \((4 n^{5})^{3}\), we start by applying the power of a product rule, which states that \((ab)^{m} = a^{m} b^{m}\). Here, we have \(a = 4\) and \(b = n^{5}\) with \(m = 3\). First, we apply the exponent to the constant: \[ 4^{3} = 64 \] Next, we apply the exponent to \(n^{5}\): \[ (n^{5})^{3} = n^{5 \cdot 3} = n^{15} \] Combining these results, we find: \[ (4 n^{5})^{3} = 64 n^{15} \] So the final simplified expression is: \[ \boxed{64 n^{15}} \]