1. **Problem:** Simplify the logarithmic expression $$\frac{2\sqrt{x} \cdot y^2}{3 \sqrt[3]{x}^4}$$. 2. **Recall the rules:** - Square root: $$\sqrt{x} = x^{\frac{1}{2}}$$ - Cube root: $$\sqrt[3]{x} = x^{\frac{1}{3}}$$ - Powers of powers: $$\left(x^a\right)^b = x^{ab}$$ - Division of powers with the same base: $$\frac{x^m}{x^n} = x^{m-n}$$ 3. **Rewrite the expression using exponents:** $$\frac{2 x^{\frac{1}{2}} y^2}{3 x^{\frac{4}{3}}}$$ 4. **Combine the powers of $$x$$ in numerator and denominator:** $$= \frac{2 y^2}{3} \cdot \frac{x^{\frac{1}{2}}}{x^{\frac{4}{3}}}$$ 5. **Subtract exponents of $$x$$:** $$= \frac{2 y^2}{3} \cdot x^{\frac{1}{2} - \frac{4}{3}}$$ 6. **Calculate the exponent difference:** $$\frac{1}{2} - \frac{4}{3} = \frac{3}{6} - \frac{8}{6} = -\frac{5}{6}$$ 7. **Final simplified expression:** $$= \frac{2 y^2}{3} x^{-\frac{5}{6}} = \frac{2 y^2}{3 x^{\frac{5}{6}}}$$ **Answer:** $$\boxed{\frac{2 y^2}{3 x^{\frac{5}{6}}}}$$