1. **State the problem:** Simplify the expression $\left(3x^{-2} y\right)^2$. 2. **Recall the power of a product rule:** When raising a product to a power, raise each factor to that power: $$\left(ab\right)^n = a^n b^n$$ 3. **Apply the rule:** $$\left(3x^{-2} y\right)^2 = 3^2 \cdot \left(x^{-2}\right)^2 \cdot y^2$$ 4. **Simplify each term:** - $3^2 = 9$ - $\left(x^{-2}\right)^2 = x^{-4}$ (multiply exponents: $-2 \times 2 = -4$) - $y^2$ remains as is 5. **Combine all:** $$9 x^{-4} y^2$$ 6. **Interpretation:** The expression simplifies to $9 x^{-4} y^2$, which matches the given expression. **Final answer:** $$9 x^{-4} y^2$$