Multiply the rational expressions. \( \frac{32 w^{5}}{27} \cdot \frac{3}{2 w^{7}}=\square \)

To multiply the rational expressions: \[ \frac{32 w^{5}}{27} \cdot \frac{3}{2 w^{7}} \] **Step 1: Multiply the coefficients and the variables separately.** - **Coefficients:** \[ \frac{32}{27} \times \frac{3}{2} = \frac{32 \times 3}{27 \times 2} = \frac{96}{54} \] Simplify \(\frac{96}{54}\) by dividing both the numerator and the denominator by 6: \[ \frac{96 \div 6}{54 \div 6} = \frac{16}{9} \] - **Variables:** \[ w^{5} \times \frac{1}{w^{7}} = \frac{w^{5}}{w^{7}} = w^{5-7} = w^{-2} = \frac{1}{w^2} \] **Step 2: Combine the simplified coefficients and variables.** \[ \frac{16}{9} \times \frac{1}{w^2} = \frac{16}{9 w^2} \] **Final Answer:** \[ \frac{16}{9 w^{2}} \]