1. **State the problem:** Simplify the expression $\left(x^{-\frac{1}{3}} \cdot y^{-\frac{1}{2}}\right)^9 \cdot \sqrt{y}$.\n\n2. **Recall the rules:** When raising a product to a power, raise each factor to that power: $\left(ab\right)^n = a^n b^n$. Also, $\sqrt{y} = y^{\frac{1}{2}}$.\n\n3. **Apply the power to each factor:**\n$$\left(x^{-\frac{1}{3}} \cdot y^{-\frac{1}{2}}\right)^9 = x^{-\frac{1}{3} \cdot 9} \cdot y^{-\frac{1}{2} \cdot 9} = x^{-3} \cdot y^{-\frac{9}{2}}$$\n\n4. **Rewrite the entire expression:**\n$$x^{-3} \cdot y^{-\frac{9}{2}} \cdot y^{\frac{1}{2}}$$\n\n5. **Combine the powers of $y$ by adding exponents:**\n$$y^{-\frac{9}{2} + \frac{1}{2}} = y^{-\frac{8}{2}} = y^{-4}$$\n\n6. **Final simplified expression:**\n$$x^{-3} \cdot y^{-4} = \frac{1}{x^3 y^4}$$\n\n**Answer:** $\frac{1}{x^3 y^4}$