1. **State the problem:** Simplify the expression $$\frac{(-3b^4)^3}{3b^8h^{-9}}$$. 2. **Apply the power to the numerator:** Use the rule $$(ab)^n = a^n b^n$$ to expand the numerator. $$(-3b^4)^3 = (-3)^3 (b^4)^3 = -27 b^{12}$$ 3. **Rewrite the expression:** $$\frac{-27 b^{12}}{3 b^8 h^{-9}}$$ 4. **Divide the coefficients:** $$\frac{-27}{3} = -9$$ 5. **Divide the powers of $b$ using the rule $\frac{b^m}{b^n} = b^{m-n}$:** $$b^{12-8} = b^4$$ 6. **Rewrite the expression with $h^{-9}$ in the denominator:** $$\frac{-9 b^4}{h^{-9}}$$ 7. **Use the rule $\frac{1}{h^{-9}} = h^9$ to move $h^{-9}$ to the numerator:** $$-9 b^4 h^9$$ **Final simplified expression:** $$\boxed{-9 b^4 h^9}$$