1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{x^3}{\sqrt{x}}$$ 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:** Recall that $$\sqrt{x} = x^{\frac{1}{2}}$$, so we can rewrite $$f(x)$$ as: $$f(x) = \frac{x^3}{x^{\frac{1}{2}}} = x^{3 - \frac{1}{2}} = x^{\frac{6}{2} - \frac{1}{2}} = x^{\frac{5}{2}}$$ 3. **Apply the power rule for derivatives:** The power rule states that $$\frac{d}{dx} x^m = m x^{m-1}$$ for any real number $$m$$. 4. **Differentiate:** Using the power rule, $$f'(x) = \frac{d}{dx} x^{\frac{5}{2}} = \frac{5}{2} x^{\frac{5}{2} - 1} = \frac{5}{2} x^{\frac{3}{2}}$$ 5. **Identify the exponent $$n$$:** From the derivative expression $$f'(x) = a x^n$$, we see that $$n = \frac{3}{2}$$. **Final answer:** $$n = \frac{3}{2}$$