1. **State the problem:** Simplify and solve the expression $$\left(\frac{a^4 b^2}{a^{-1} b^3}\right)^{-2}$$. 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: $$(x^m)^n = x^{mn}$$. - Negative exponents mean reciprocal: $$x^{-m} = \frac{1}{x^m}$$. 3. **Simplify inside the parentheses:** $$\frac{a^4 b^2}{a^{-1} b^3} = a^{4 - (-1)} b^{2 - 3} = a^{4 + 1} b^{-1} = a^5 b^{-1}$$ 4. **Apply the outer exponent -2:** $$\left(a^5 b^{-1}\right)^{-2} = a^{5 \times (-2)} b^{-1 \times (-2)} = a^{-10} b^{2}$$ 5. **Express with positive exponents:** $$a^{-10} b^{2} = \frac{b^2}{a^{10}}$$ **Final answer:** $$\boxed{\frac{b^2}{a^{10}}}$$