1. **State the problem:** Simplify the expression $$\frac{8r^4 s^9}{4r^3 s^3}$$. 2. **Recall the rules:** When dividing powers with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. 3. **Apply the division to coefficients:** $$\frac{8}{4} = \cancel{\frac{8}{4}} = 2$$. 4. **Apply the exponent subtraction for $r$:** $$\frac{r^4}{r^3} = r^{4-3} = r^1 = r$$. 5. **Apply the exponent subtraction for $s$:** $$\frac{s^9}{s^3} = s^{9-3} = s^6$$. 6. **Combine all parts:** $$2 \times r \times s^6 = 2r s^6$$. **Final answer:** $$2r s^6$$