1. **Problem:** Simplify $$\frac{(a^4 b^{-3})^5}{a^{-7} b^8}$$ and express your answer with positive indices. 2. **Formula and rules:** - Power of a power: $$(x^m)^n = x^{mn}$$ - Division of powers with the same base: $$\frac{x^a}{x^b} = x^{a-b}$$ - Negative exponent rule: $$x^{-m} = \frac{1}{x^m}$$ 3. **Step-by-step simplification:** - Apply power of a power: $$(a^4 b^{-3})^5 = a^{4 \times 5} b^{-3 \times 5} = a^{20} b^{-15}$$ - Substitute into the original expression: $$\frac{a^{20} b^{-15}}{a^{-7} b^8}$$ - Use division of powers: $$a^{20 - (-7)} b^{-15 - 8} = a^{20 + 7} b^{-23} = a^{27} b^{-23}$$ - Express with positive indices: $$a^{27} \times \frac{1}{b^{23}} = \frac{a^{27}}{b^{23}}$$ **Final answer:** $$\frac{a^{27}}{b^{23}}$$