1. **State the problem:** We need to find the derivative of the function $$f(x) = \sqrt[3]{x^4}$$ and express it in the form $$f'(x) = ax^n$$, then find the exact value of the exponent $$n$$ as a simplified fraction. 2. **Rewrite the function using exponents:** Recall that $$\sqrt[3]{x^4} = x^{\frac{4}{3}}$$. 3. **Use the power rule for derivatives:** The derivative of $$x^m$$ is $$mx^{m-1}$$. 4. **Apply the power rule:** $$f'(x) = \frac{4}{3} x^{\frac{4}{3} - 1} = \frac{4}{3} x^{\frac{4}{3} - \frac{3}{3}} = \frac{4}{3} x^{\frac{1}{3}}$$ 5. **Identify the exponent $$n$$:** The exponent $$n$$ in $$f'(x) = ax^n$$ is $$\frac{1}{3}$$. 6. **Final answer:** $$n = \frac{1}{3}$$.