Simplify. \[ \frac{\left(4 x^{4}\right)^{2}}{\left(3 x^{5}\right)^{3}} \]

To simplify the expression \(\frac{\left(4 x^{4}\right)^{2}}{\left(3 x^{5}\right)^{3}}\), let's break it down step by step. First, we simplify the numerator \((4 x^{4})^{2}\): \[ (4 x^{4})^{2} = 4^{2} \cdot (x^{4})^{2} = 16 \cdot x^{8} = 16 x^{8}. \] Next, we simplify the denominator \((3 x^{5})^{3}\): \[ (3 x^{5})^{3} = 3^{3} \cdot (x^{5})^{3} = 27 \cdot x^{15} = 27 x^{15}. \] Now we can rewrite the entire expression: \[ \frac{16 x^{8}}{27 x^{15}}. \] To simplify further, we divide the coefficients and the powers of \(x\): \[ = \frac{16}{27} \cdot \frac{x^{8}}{x^{15}}. \] When dividing powers with the same base, we subtract the exponents: \[ \frac{x^{8}}{x^{15}} = x^{8 - 15} = x^{-7}. \] Putting it all together, we have: \[ \frac{16}{27} x^{-7}. \] We can express \(x^{-7}\) as \(\frac{1}{x^{7}}\): \[ = \frac{16}{27} \cdot \frac{1}{x^{7}} = \frac{16}{27 x^{7}}. \] Thus, the simplified expression is: \[ \frac{16}{27 x^{7}}. \]