1. **State the problem:** Simplify the expression $$\left(\frac{x^{-2}}{x y^{2}}\right)^3$$. 2. **Recall the rules:** - When dividing powers with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. - When raising a power to another power, multiply the exponents: $$(a^m)^n = a^{m \cdot n}$$. - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^m}$$. 3. **Simplify inside the parentheses first:** $$\frac{x^{-2}}{x y^{2}} = x^{-2} \cdot \frac{1}{x y^{2}} = x^{-2} \cdot x^{-1} \cdot y^{-2} = x^{-3} y^{-2}$$ 4. **Apply the exponent 3 to each factor:** $$\left(x^{-3} y^{-2}\right)^3 = x^{-3 \cdot 3} y^{-2 \cdot 3} = x^{-9} y^{-6}$$ 5. **Rewrite with positive exponents:** $$x^{-9} y^{-6} = \frac{1}{x^{9} y^{6}}$$ **Final answer:** $$\boxed{\frac{1}{x^{9} y^{6}}}$$