1. **State the problem:** Simplify the expression $$\left(\frac{y}{x^2}\right)^3 \left(\frac{x}{y^2}\right)^2$$. 2. **Recall the power of a quotient rule:** For any nonzero $a$ and $b$, and integer $n$, $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$. 3. **Apply the rule to each factor:** $$\left(\frac{y}{x^2}\right)^3 = \frac{y^3}{(x^2)^3} = \frac{y^3}{x^6}$$ $$\left(\frac{x}{y^2}\right)^2 = \frac{x^2}{(y^2)^2} = \frac{x^2}{y^4}$$ 4. **Multiply the two results:** $$\frac{y^3}{x^6} \times \frac{x^2}{y^4} = \frac{y^3 \cdot x^2}{x^6 \cdot y^4}$$ 5. **Combine like bases by subtracting exponents:** $$= \frac{x^2}{x^6} \times \frac{y^3}{y^4} = x^{2-6} y^{3-4} = x^{-4} y^{-1}$$ 6. **Rewrite with positive exponents:** $$x^{-4} y^{-1} = \frac{1}{x^4 y}$$ **Final answer:** $$\boxed{\frac{1}{x^4 y}}$$