1. **State the problem:** Simplify the expression $$\frac{x^3}{x^{1/2}}$$ and find the value of $a$ such that $$\frac{x^3}{x^{1/2}} = x^a$$. 2. **Recall the exponent rule:** When dividing powers with the same base, subtract the exponents: $$x^m \div x^n = x^{m-n}$$ 3. **Apply the rule:** $$\frac{x^3}{x^{1/2}} = x^{3 - \frac{1}{2}}$$ 4. **Simplify the exponent:** $$3 - \frac{1}{2} = \frac{6}{2} - \frac{1}{2} = \frac{5}{2}$$ 5. **Final expression:** $$x^a = x^{\frac{5}{2}}$$ 6. **Answer:** $$a = \frac{5}{2}$$