1. **State the problem:** Simplify the expression $$\left( \frac{4x^{-3} y^{4}}{8x^{2} y^{-2}} \right)^{-2}$$. 2. **Apply the quotient rule for exponents:** When dividing like bases, subtract the exponents. Inside the parentheses, simplify the fraction: $$\frac{4x^{-3} y^{4}}{8x^{2} y^{-2}} = \frac{4}{8} \cdot x^{-3 - 2} \cdot y^{4 - (-2)} = \frac{4}{8} \cdot x^{-5} \cdot y^{6}$$ 3. **Simplify the coefficient:** $$\frac{4}{8} = \frac{\cancel{4}}{\cancel{8}} = \frac{1}{2}$$ So the expression inside the parentheses is: $$\frac{1}{2} x^{-5} y^{6}$$ 4. **Apply the negative exponent outside the parentheses:** $$\left( \frac{1}{2} x^{-5} y^{6} \right)^{-2} = \left( \frac{1}{2} \right)^{-2} \cdot (x^{-5})^{-2} \cdot (y^{6})^{-2}$$ 5. **Simplify each part:** - $$\left( \frac{1}{2} \right)^{-2} = 2^{2} = 4$$ - $$(x^{-5})^{-2} = x^{(-5) \times (-2)} = x^{10}$$ - $$(y^{6})^{-2} = y^{6 \times (-2)} = y^{-12}$$ 6. **Combine all parts:** $$4 x^{10} y^{-12}$$ 7. **Rewrite with positive exponents only:** $$4 x^{10} \frac{1}{y^{12}} = \frac{4 x^{10}}{y^{12}}$$ **Final answer:** $$\boxed{\frac{4 x^{10}}{y^{12}}}$$