1. **State the problem:** Simplify the expression $$\left(\frac{4a^{-2} b^{3}}{8a^{4} b^{-4}}\right)^{-2}$$. 2. **Recall the rules:** - When dividing powers with the same base, subtract exponents: $$a^{m} / a^{n} = a^{m-n}$$. - When raising a power to another power, multiply exponents: $$(x^{m})^{n} = x^{mn}$$. - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^{m}}$$. 3. **Simplify inside the parentheses first:** $$\frac{4a^{-2} b^{3}}{8a^{4} b^{-4}} = \frac{4}{8} \cdot a^{-2-4} \cdot b^{3-(-4)} = \frac{1}{2} \cdot a^{-6} \cdot b^{7}$$ 4. **Rewrite the expression:** $$\left(\frac{1}{2} a^{-6} b^{7}\right)^{-2}$$ 5. **Apply the power of -2 to each factor:** $$\left(\frac{1}{2}\right)^{-2} \cdot (a^{-6})^{-2} \cdot (b^{7})^{-2} = 2^{2} \cdot a^{12} \cdot b^{-14}$$ 6. **Calculate powers:** $$2^{2} = 4$$ 7. **Final simplified expression:** $$4 a^{12} b^{-14} = \frac{4 a^{12}}{b^{14}}$$ **Answer:** $$\boxed{\frac{4 a^{12}}{b^{14}}}$$