(2/3ab²)³3/2a³b²

To simplify the expression \((\frac{2}{3}ab^2)^{\frac{3}{2}} \cdot \frac{3}{2}a^3b^2\), we can follow these steps: 1. Simplify the first part \((\frac{2}{3}ab^2)^{\frac{3}{2}}\): \[ (\frac{2}{3})^{\frac{3}{2}} \cdot a^{\frac{3}{2}} \cdot (b^2)^{\frac{3}{2}} \] Calculating each part: - \((\frac{2}{3})^{\frac{3}{2}} = \frac{2^{\frac{3}{2}}}{3^{\frac{3}{2}}} = \frac{2\sqrt{2}}{3\sqrt{3}}\) - \((b^2)^{\frac{3}{2}} = b^{3}\) Thus, \[ (\frac{2}{3}ab^2)^{\frac{3}{2}} = \frac{2\sqrt{2}}{3\sqrt{3}} \cdot a^{\frac{3}{2}} \cdot b^3 \] 2. Now, we multiply this result by \(\frac{3}{2}a^3b^2\): \[ \frac{2\sqrt{2}}{3\sqrt{3}} \cdot a^{\frac{3}{2}} \cdot b^3 \cdot \frac{3}{2}a^3b^2 \] 3. Combine the coefficients and the variables: - Coefficients: \[ \frac{2\sqrt{2}}{3\sqrt{3}} \cdot \frac{3}{2} = \sqrt{2}/\sqrt{3} \] - For \(a\): \[ a^{\frac{3}{2}} \cdot a^3 = a^{\frac{3}{2} + 3} = a^{\frac{3}{2} + \frac{6}{2}} = a^{\frac{9}{2}} \] - For \(b\): \[ b^3 \cdot b^2 = b^{3 + 2} = b^5 \] Putting it all together, we have: \[ \frac{\sqrt{2}}{\sqrt{3}} \cdot a^{\frac{9}{2}} \cdot b^5 \] So, the simplified expression is: \[ \frac{\sqrt{2}}{\sqrt{3}} a^{\frac{9}{2}} b^5 \]