1. **State the problem:** Simplify the expression $$\frac{(2p^{m-1} q)^{-4} \cdot 2m^{-1} p^3}{2pq^2}$$ and verify if the given process is correct. 2. **Recall the rules:** - When raising a product to a power, apply the power to each factor: $$(ab)^n = a^n b^n$$ - Negative exponents mean reciprocal: $$a^{-n} = \frac{1}{a^n}$$ - When dividing like bases, subtract exponents: $$\frac{a^m}{a^n} = a^{m-n}$$ 3. **Simplify the numerator:** $$(2p^{m-1} q)^{-4} \cdot 2m^{-1} p^3 = 2^{-4} p^{-4(m-1)} q^{-4} \cdot 2 m^{-1} p^3$$ 4. **Combine constants and like bases:** $$2^{-4} \cdot 2 = \cancel{2^{-4+1}} = 2^{-3}$$ $$p^{-4(m-1)} \cdot p^3 = p^{-4m + 4 + 3} = p^{-4m + 7}$$ 5. **Rewrite numerator:** $$2^{-3} m^{-1} p^{-4m + 7} q^{-4}$$ 6. **Divide by denominator:** $$\frac{2^{-3} m^{-1} p^{-4m + 7} q^{-4}}{2 p q^2} = 2^{-3} m^{-1} p^{-4m + 7} q^{-4} \cdot \frac{1}{2 p q^2}$$ 7. **Simplify constants and variables:** $$2^{-3} \cdot \frac{1}{2} = 2^{-3 - 1} = 2^{-4}$$ $$p^{-4m + 7} \cdot p^{-1} = p^{-4m + 6}$$ $$q^{-4} \cdot q^{-2} = q^{-6}$$ 8. **Final simplified expression:** $$\boxed{\frac{p^{-4m + 6}}{2^{4} m q^{6}}}$$ This matches the correct application of exponent rules. Your process is mostly correct if it follows these steps.