1. **State the problem:** Simplify the expression $$\left(\frac{a^3 b^{-2}}{a^{-1} b^{5/2}}\right)^2$$. 2. **Recall the rules:** - When dividing powers with the same base, subtract exponents: $$\frac{x^m}{x^n} = x^{m-n}$$. - When raising a power to another power, multiply exponents: $$(x^m)^n = x^{mn}$$. 3. **Simplify inside the parentheses:** $$\frac{a^3 b^{-2}}{a^{-1} b^{5/2}} = a^{3 - (-1)} b^{-2 - \frac{5}{2}} = a^{3 + 1} b^{-2 - 2.5} = a^4 b^{-\frac{9}{2}}$$ 4. **Apply the outer exponent 2:** $$\left(a^4 b^{-\frac{9}{2}}\right)^2 = a^{4 \times 2} b^{-\frac{9}{2} \times 2} = a^8 b^{-9}$$ 5. **Final answer:** $$a^8 b^{-9} = \frac{a^8}{b^9}$$