1. **State the problem:** Simplify the expression $$\frac{(3xy^{-2})^{-2}}{3x^{-2}y}$$. 2. **Recall exponent rules:** - Power of a power: $$(a^m)^n = a^{mn}$$ - Negative exponent: $$a^{-m} = \frac{1}{a^m}$$ - Product of powers: $$a^m \cdot a^n = a^{m+n}$$ - Quotient of powers: $$\frac{a^m}{a^n} = a^{m-n}$$ 3. **Simplify numerator:** $$(3xy^{-2})^{-2} = 3^{-2} \cdot x^{-2} \cdot (y^{-2})^{-2} = 3^{-2} x^{-2} y^{4}$$ 4. **Rewrite the expression:** $$\frac{3^{-2} x^{-2} y^{4}}{3 x^{-2} y}$$ 5. **Divide coefficients and variables:** $$= 3^{-2 - 1} \cdot x^{-2 - (-2)} \cdot y^{4 - 1}$$ 6. **Simplify exponents:** $$= 3^{-3} \cdot x^{0} \cdot y^{3}$$ 7. **Simplify further:** Since $$x^{0} = 1$$, $$= \frac{y^{3}}{3^{3}} = \frac{y^{3}}{27}$$ **Final answer:** $$\frac{y^{3}}{27}$$