1. Start by stating the problem: Simplify the expression $$\frac{(4a^3b^2)^2}{(a^2b)^2}$$. 2. Apply the power of a product rule to both numerator and denominator: $$ (4a^3b^2)^2 = 4^2 (a^3)^2 (b^2)^2 = 16 a^{6} b^{4} $$ $$ (a^2 b)^2 = (a^2)^2 (b)^2 = a^{4} b^{2} $$ 3. Substitute these back into the expression: $$ \frac{16 a^{6} b^{4}}{a^{4} b^{2}} $$ 4. Simplify by canceling terms with the same base using the rule $$ \frac{x^m}{x^n} = x^{m-n} $$: $$ 16 a^{6-4} b^{4-2} = 16 a^{2} b^{2} $$ 5. Final simplified expression is: $$ 16 a^{2} b^{2} $$