1. **State the problem:** Simplify the expression $$\frac{48a^3 b^2}{18a b^5}$$. 2. **Write the formula and rules:** When simplifying fractions with variables, divide coefficients and subtract exponents of like bases using the rule $$\frac{x^m}{x^n} = x^{m-n}$$. 3. **Simplify the coefficients:** $$\frac{48}{18}$$ $$\frac{\cancel{6} \times 8}{\cancel{6} \times 3}$$ $$\frac{8}{3}$$ 4. **Simplify the variable $a$ terms:** $$\frac{a^3}{a}$$ $$\frac{\cancel{a} \times a^2}{\cancel{a}}$$ $$a^{3-1} = a^2$$ 5. **Simplify the variable $b$ terms:** $$\frac{b^2}{b^5}$$ $$\frac{b^2}{\cancel{b^2} \times b^3}$$ $$b^{2-5} = b^{-3} = \frac{1}{b^3}$$ 6. **Combine all simplified parts:** $$\frac{8}{3} \times a^2 \times \frac{1}{b^3} = \frac{8a^2}{3b^3}$$ **Final answer:** $$\frac{8a^2}{3b^3}$$