1. **State the problem:** Simplify the expression $$\left(\frac{-12x^{-5}y}{4x^{3}}\right)^{-2}$$. 2. **Recall the rules:** - When raising a fraction to a power, raise both numerator and denominator to that power: $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$. - When raising a power to another power, multiply exponents: $$(x^a)^b = x^{ab}$$. - Negative exponents mean reciprocal: $$x^{-a} = \frac{1}{x^a}$$. 3. **Simplify inside the parentheses first:** $$\frac{-12x^{-5}y}{4x^{3}} = \frac{-12}{4} \cdot \frac{x^{-5}}{x^{3}} \cdot y = -3 \cdot x^{-5-3} \cdot y = -3x^{-8}y$$ 4. **Rewrite the expression:** $$\left(-3x^{-8}y\right)^{-2}$$ 5. **Apply the power of -2:** $$(-3)^{-2} \cdot (x^{-8})^{-2} \cdot y^{-2}$$ 6. **Simplify each part:** - $$(-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}$$ - $$(x^{-8})^{-2} = x^{16}$$ - $$y^{-2} = \frac{1}{y^2}$$ 7. **Combine all parts:** $$\frac{1}{9} \cdot x^{16} \cdot \frac{1}{y^2} = \frac{x^{16}}{9y^{2}}$$ **Final answer:** $$\boxed{\frac{x^{16}}{9y^{2}}}$$