1. **Stating the problem:** Solve the equation $x^{\frac{1}{3}} = 3$ for $x$. 2. **Formula and rules:** The equation involves a cube root. Recall that $x^{\frac{1}{3}}$ means the cube root of $x$, so to solve for $x$, we raise both sides of the equation to the power of 3 to undo the cube root. 3. **Intermediate work:** $$\left(x^{\frac{1}{3}}\right)^3 = 3^3$$ Using the power of a power rule, $\left(a^{m}\right)^n = a^{mn}$, we get: $$x^{\frac{1}{3} \times 3} = 27$$ $$x^1 = 27$$ $$x = 27$$ 4. **Explanation:** Raising both sides to the power of 3 cancels the cube root on the left side, leaving $x$ alone. The right side becomes $3^3 = 27$. So the solution is $x = 27$. **Final answer:** $x = 27$