1. **State the problem:** Multiply and simplify the expression $$\frac{9m^2 n}{12m n^4} \times \frac{16m^3 n^2}{6m^2}$$. 2. **Write the multiplication of fractions:** $$\frac{9m^2 n}{12m n^4} \times \frac{16m^3 n^2}{6m^2} = \frac{9m^2 n \times 16m^3 n^2}{12m n^4 \times 6m^2}$$ 3. **Multiply numerators and denominators:** Numerator: $$9 \times 16 \times m^{2+3} \times n^{1+2} = 144 m^5 n^3$$ Denominator: $$12 \times 6 \times m^{1+2} \times n^4 = 72 m^3 n^4$$ 4. **Simplify the coefficients:** $$\frac{144}{72} = 2$$ 5. **Simplify the variables using the rule $$\frac{a^x}{a^y} = a^{x-y}$$:** $$m^{5-3} = m^2$$ $$n^{3-4} = n^{-1} = \frac{1}{n}$$ 6. **Combine all simplified parts:** $$2 m^2 \times \frac{1}{n} = \frac{2 m^2}{n}$$ **Final answer:** $$\frac{2 m^2}{n}$$