1. **Problem Statement:** Simplify the expression $$\frac{\sqrt[3]{x^6 y^9}}{(x^{\frac{1}{2}} y^{\frac{1}{3}})^3} \cdot \sqrt{x^4 y^2}$$ where $x > 0$ and $y > 0$. 2. **Recall the laws of exponents and radicals:** - $\sqrt[n]{a^m} = a^{\frac{m}{n}}$ - $(a^m)^n = a^{mn}$ - $\frac{a^m}{a^n} = a^{m-n}$ - $a^m \cdot a^n = a^{m+n}$ 3. **Rewrite each radical and power using fractional exponents:** - $\sqrt[3]{x^6 y^9} = (x^6 y^9)^{\frac{1}{3}} = x^{6 \cdot \frac{1}{3}} y^{9 \cdot \frac{1}{3}} = x^2 y^3$ - $(x^{\frac{1}{2}} y^{\frac{1}{3}})^3 = x^{\frac{1}{2} \cdot 3} y^{\frac{1}{3} \cdot 3} = x^{\frac{3}{2}} y^1$ - $\sqrt{x^4 y^2} = (x^4 y^2)^{\frac{1}{2}} = x^{4 \cdot \frac{1}{2}} y^{2 \cdot \frac{1}{2}} = x^2 y^1$ 4. **Substitute back into the expression:** $$\frac{x^2 y^3}{x^{\frac{3}{2}} y^1} \cdot x^2 y^1$$ 5. **Simplify the fraction using exponent subtraction:** $$x^{2 - \frac{3}{2}} y^{3 - 1} = x^{\frac{1}{2}} y^2$$ 6. **Multiply by the last term:** $$x^{\frac{1}{2}} y^2 \cdot x^2 y^1 = x^{\frac{1}{2} + 2} y^{2 + 1} = x^{\frac{5}{2}} y^3$$ 7. **Final simplified expression:** $$\boxed{x^{\frac{5}{2}} y^3}$$ This expression follows all the laws of exponents and radicals, demonstrating the use of fractional exponents, multiplication, division, and simplification.