1. **State the problem:** Simplify the expression $$\ln\left(\frac{\sqrt[3]{x^2}}{w+z}\right)$$. 2. **Recall the logarithm property:** For any positive values, $$\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$$. 3. **Apply the property:** $$\ln\left(\frac{\sqrt[3]{x^2}}{w+z}\right) = \ln\left(\sqrt[3]{x^2}\right) - \ln(w+z)$$ 4. **Rewrite the cube root as a fractional exponent:** $$\sqrt[3]{x^2} = x^{\frac{2}{3}}$$ 5. **Use the logarithm power rule:** $$\ln\left(x^{\frac{2}{3}}\right) = \frac{2}{3} \ln(x)$$ 6. **Combine the results:** $$\ln\left(\frac{\sqrt[3]{x^2}}{w+z}\right) = \frac{2}{3} \ln(x) - \ln(w+z)$$ **Final answer:** $$\boxed{\frac{2}{3} \ln(x) - \ln(w+z)}$$