Enter the simplified exponential form of the following expression. Use only positive exponents in your simplified expression. \[ \left(\frac{3 x^{-4} y^{3}}{12 x y^{-4}}\right)^{-2} \]

To simplify the given expression \(\left(\frac{3 x^{-4} y^{3}}{12 x y^{-4}}\right)^{-2}\), we start by simplifying the fraction inside the parentheses: 1. Simplify the coefficients: \(\frac{3}{12} = \frac{1}{4}\). 2. For the \(x\) terms: \[ x^{-4} \text{ in the numerator and } x^{1} \text{ in the denominator gives } x^{-4 - 1} = x^{-5}. \] 3. For the \(y\) terms: \[ y^{3} \text{ in the numerator and } y^{-4} \text{ in the denominator gives } y^{3 - (-4)} = y^{3 + 4} = y^{7}. \] Now, combine everything together: \[ \frac{3 x^{-4} y^{3}}{12 x y^{-4}} = \frac{1}{4} \cdot \frac{y^{7}}{x^{5}} = \frac{y^{7}}{4 x^{5}}. \] Next, apply the exponent of \(-2\): \[ \left(\frac{y^{7}}{4 x^{5}}\right)^{-2} = \frac{1}{\left(\frac{y^{7}}{4 x^{5}}\right)^{2}} = \frac{(4 x^{5})^{2}}{(y^{7})^{2}} = \frac{16 x^{10}}{y^{14}}. \] Thus, the simplified expression with only positive exponents is: \[ \frac{16 x^{10}}{y^{14}}. \]