1. **State the problem:** Solve the equation $$x^{\frac{2}{3}} - 2x^{\frac{1}{3}} - 3 = 0$$ for $x$. 2. **Use substitution:** Let $$y = x^{\frac{1}{3}}$$. Then $$y^2 = x^{\frac{2}{3}}$$. 3. **Rewrite the equation:** Substitute into the original equation: $$y^2 - 2y - 3 = 0$$ 4. **Solve the quadratic equation:** Use the quadratic formula or factorization. Factorization: $$(y - 3)(y + 1) = 0$$ So, $$y = 3$$ or $$y = -1$$. 5. **Back-substitute for $x$:** Recall $$y = x^{\frac{1}{3}}$$, so - If $$y = 3$$, then $$x^{\frac{1}{3}} = 3 \implies x = 3^3 = 27$$. - If $$y = -1$$, then $$x^{\frac{1}{3}} = -1 \implies x = (-1)^3 = -1$$. 6. **Final answer:** $$x = 27, -1$$. These are the simplified real solutions to the equation.