1. **State the problem:** Simplify the expression $$\frac{48a^3 b^2}{18a b^5}$$. 2. **Recall the rules:** - When dividing powers with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. - Simplify coefficients by dividing their greatest common divisor. 3. **Simplify the coefficients:** $$\frac{48}{18} = \frac{48 \div 6}{18 \div 6} = \frac{8}{3}$$ 4. **Simplify the variables:** $$\frac{a^3}{a} = a^{3-1} = a^2$$ $$\frac{b^2}{b^5} = b^{2-5} = b^{-3} = \frac{1}{b^3}$$ 5. **Combine all parts:** $$\frac{48a^3 b^2}{18a b^5} = \frac{8}{3} \times a^2 \times \frac{1}{b^3} = \frac{8a^2}{3b^3}$$ **Final answer:** $$\frac{8a^2}{3b^3}$$