1. **State the problem:** Simplify the expression $$\frac{a^2bc^3}{ab^2c^{-1}}$$. 2. **Recall the rules:** When dividing powers with the same base, subtract the exponents: $$\frac{x^m}{x^n} = x^{m-n}$$. 3. **Apply the rule to each variable:** $$\frac{a^2}{a} = a^{2-1} = a^1 = a$$ $$\frac{b}{b^2} = b^{1-2} = b^{-1}$$ $$\frac{c^3}{c^{-1}} = c^{3 - (-1)} = c^{3+1} = c^4$$ 4. **Combine the results:** $$a \times b^{-1} \times c^4 = \frac{a c^4}{b}$$ 5. **Final answer:** $$\boxed{\frac{a c^4}{b}}$$