1. **State the problem:** Simplify the expression $$3.16a^5 b^{-3} - 8a^3 b^{-5} + 24a^8 b^{-7} - 40a^2 b^{-9}$$. 2. **Recall the rules:** When simplifying expressions with variables raised to powers, use the laws of exponents: - $a^m \cdot a^n = a^{m+n}$ - $\frac{a^m}{a^n} = a^{m-n}$ - $a^{-n} = \frac{1}{a^n}$ 3. **Rewrite the expression to factor common terms:** Look for the smallest powers of $a$ and $b$ in all terms: - Smallest power of $a$ is $a^2$ - Smallest power of $b$ is $b^{-9}$ 4. **Factor out $a^2 b^{-9}$:** $$ 3.16a^5 b^{-3} - 8a^3 b^{-5} + 24a^8 b^{-7} - 40a^2 b^{-9} = a^2 b^{-9} \left(3.16a^{5-2} b^{-3+9} - 8a^{3-2} b^{-5+9} + 24a^{8-2} b^{-7+9} - 40\right) $$ 5. **Simplify the exponents inside the parentheses:** $$ = a^2 b^{-9} \left(3.16a^3 b^6 - 8a^1 b^4 + 24a^6 b^2 - 40\right) $$ 6. **Final simplified form:** $$ \boxed{a^2 b^{-9} \left(3.16a^3 b^6 - 8a b^4 + 24a^6 b^2 - 40\right)} $$ This is the factored form showing the common factors extracted for clarity and simplification.