1. **Stating the problem:** Simplify the expression $$\left( \frac{a^2 b^{-3}}{a^{-1} b^{\frac{3}{2}}} \right)^{\frac{3}{2}}$$. 2. **Recall the rules:** - When dividing powers with the same base, subtract the exponents: $$\frac{x^m}{x^n} = x^{m-n}$$. - When raising a power to another power, multiply the exponents: $$(x^m)^n = x^{mn}$$. 3. **Simplify inside the parentheses first:** $$\frac{a^2 b^{-3}}{a^{-1} b^{\frac{3}{2}}} = a^{2 - (-1)} b^{-3 - \frac{3}{2}} = a^{2 + 1} b^{-3 - 1.5} = a^3 b^{-4.5}$$ 4. **Rewrite the expression:** $$\left(a^3 b^{-4.5}\right)^{\frac{3}{2}}$$ 5. **Apply the power to each factor:** $$a^{3 \times \frac{3}{2}} b^{-4.5 \times \frac{3}{2}} = a^{\frac{9}{2}} b^{-\frac{27}{4}}$$ 6. **Final simplified form:** $$a^{\frac{9}{2}} b^{-\frac{27}{4}}$$ This is the simplest form of the given expression.