1. **State the problem:** Simplify the expression $$\left(\frac{3pq^6}{p^4q}\right)^{-2}$$ and express the answer with positive exponents. 2. **Rewrite the expression inside the parentheses:** $$\frac{3pq^6}{p^4q} = 3 \cdot \frac{p}{p^4} \cdot \frac{q^6}{q}$$ 3. **Simplify each part using exponent rules:** $$\frac{p}{p^4} = p^{1-4} = p^{-3}$$ $$\frac{q^6}{q} = q^{6-1} = q^5$$ 4. **Substitute back:** $$3 \cdot p^{-3} \cdot q^5 = 3p^{-3}q^5$$ 5. **Apply the outer exponent of -2:** $$\left(3p^{-3}q^5\right)^{-2} = 3^{-2} \cdot \left(p^{-3}\right)^{-2} \cdot \left(q^5\right)^{-2}$$ 6. **Simplify each term:** $$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$$ $$\left(p^{-3}\right)^{-2} = p^{(-3) \times (-2)} = p^6$$ $$\left(q^5\right)^{-2} = q^{5 \times (-2)} = q^{-10}$$ 7. **Combine all terms:** $$\frac{1}{9} \cdot p^6 \cdot q^{-10} = \frac{p^6}{9q^{10}}$$ **Final answer:** $$\boxed{\frac{p^6}{9q^{10}}}$$