1. **State the problem:** Simplify the expression $$\frac{4p^{-2} q^{4}}{8 p^{3} (q^{-2})^{3}}$$. 2. **Recall the exponent rules:** - When dividing like bases, subtract exponents: $$a^{m} \div a^{n} = a^{m-n}$$. - Power of a power: $$(a^{m})^{n} = a^{m \cdot n}$$. 3. **Simplify the denominator's exponent:** $$(q^{-2})^{3} = q^{-2 \cdot 3} = q^{-6}$$. 4. **Rewrite the expression:** $$\frac{4p^{-2} q^{4}}{8 p^{3} q^{-6}}$$. 5. **Divide coefficients:** $$\frac{4}{8} = \frac{\cancel{4}}{\cancel{8}} = \frac{1}{2}$$. 6. **Apply exponent subtraction for $p$:** $$p^{-2} \div p^{3} = p^{-2 - 3} = p^{-5}$$. 7. **Apply exponent subtraction for $q$:** $$q^{4} \div q^{-6} = q^{4 - (-6)} = q^{4 + 6} = q^{10}$$. 8. **Combine all parts:** $$\frac{1}{2} p^{-5} q^{10} = \frac{q^{10}}{2 p^{5}}$$. **Final answer:** $$\frac{q^{10}}{2 p^{5}}$$