1. **Problem statement:** Solve the equation $$\sqrt[3]{\left( \frac{x - \frac{3}{2}}{\sqrt[3]{x + \frac{2}{3}}} \right)} = 1$$ for complex numbers. 2. **Recall the property:** For any complex number $z$, if $$\sqrt[3]{z} = 1,$$ then $$z = 1$$ because the cube root of 1 is 1 (principal root). 3. **Apply this to the equation:** $$\frac{x - \frac{3}{2}}{\sqrt[3]{x + \frac{2}{3}}} = 1$$ 4. **Rewrite the denominator:** Let $$y = \sqrt[3]{x + \frac{2}{3}}$$, then $$y^3 = x + \frac{2}{3}$$. 5. **Substitute back:** $$\frac{x - \frac{3}{2}}{y} = 1 \implies x - \frac{3}{2} = y$$ 6. **Express $x$ in terms of $y$:** $$x = y + \frac{3}{2}$$ 7. **Recall $y^3 = x + \frac{2}{3}$, substitute $x$:** $$y^3 = y + \frac{3}{2} + \frac{2}{3} = y + \frac{13}{6}$$ 8. **Rearranged cubic equation:** $$y^3 - y - \frac{13}{6} = 0$$ 9. **Solve the cubic equation:** We look for rational roots using the Rational Root Theorem. Possible roots are factors of $\frac{13}{6}$, but none satisfy the equation exactly. 10. **Use Cardano's formula or numerical methods:** The cubic has one real root and two complex roots. The real root approximately is $y \approx 1.8$ (numerical approximation). 11. **Find $x$:** $$x = y + \frac{3}{2} \approx 1.8 + 1.5 = 3.3$$ 12. **Summary:** The solution for $x$ satisfies the cubic equation derived and is approximately $x \approx 3.3$ for the principal root. **Final answer:** $$x \approx 3.3$$