1. **State the problem:** Simplify the expression $$\frac{(x^3+3x^2)^{\frac{1}{3}}}{x}$$. 2. **Recall the rules:** - The cube root of a number $a$ is written as $a^{\frac{1}{3}}$. - When dividing powers with the same base, subtract the exponents. - Factor expressions inside roots if possible to simplify. 3. **Factor the expression inside the cube root:** $$x^3 + 3x^2 = x^2(x + 3)$$ 4. **Rewrite the cube root using the factorization:** $$\left(x^2(x + 3)\right)^{\frac{1}{3}} = \left(x^2\right)^{\frac{1}{3}} \cdot (x + 3)^{\frac{1}{3}} = x^{\frac{2}{3}} (x + 3)^{\frac{1}{3}}$$ 5. **Divide by $x$:** $$\frac{x^{\frac{2}{3}} (x + 3)^{\frac{1}{3}}}{x} = x^{\frac{2}{3} - 1} (x + 3)^{\frac{1}{3}} = x^{-\frac{1}{3}} (x + 3)^{\frac{1}{3}}$$ 6. **Rewrite the negative exponent:** $$x^{-\frac{1}{3}} = \frac{1}{x^{\frac{1}{3}}}$$ 7. **Final simplified expression:** $$\frac{(x + 3)^{\frac{1}{3}}}{x^{\frac{1}{3}}} = \left(\frac{x + 3}{x}\right)^{\frac{1}{3}}$$ **Answer:** $$\boxed{\left(\frac{x + 3}{x}\right)^{\frac{1}{3}}}$$