Simplify. \[ \left(4 x^{3} y^{-4}\right)^{-2} \] Write your answer using only positive exponents.

To simplify the expression \(\left(4 x^{3} y^{-4}\right)^{-2}\), we start by applying the exponent of \(-2\) to each component inside the parentheses. This means we raise each part to the power of \(-2\): \[ \left(4\right)^{-2} \cdot \left(x^{3}\right)^{-2} \cdot \left(y^{-4}\right)^{-2} \] Calculating each term, we have: 1. \(\left(4\right)^{-2} = \frac{1}{4^{2}} = \frac{1}{16}\) 2. \(\left(x^{3}\right)^{-2} = x^{-6} = \frac{1}{x^{6}}\) 3. \(\left(y^{-4}\right)^{-2} = y^{8}\) Putting it all together, we have: \[ \left(4 x^{3} y^{-4}\right)^{-2} = \frac{1}{16} \cdot \frac{1}{x^{6}} \cdot y^{8} \] This can be rewritten as: \[ \frac{y^{8}}{16 x^{6}} \] Thus, the final simplified expression, using only positive exponents, is: \[ \frac{y^{8}}{16 x^{6}} \]