12. \( 10 a^{\frac{7}{3}} b^{-1} \cdot 3 a^{\frac{5}{3}} b^{8} \)

To simplify the expression \( 10 a^{\frac{7}{3}} b^{-1} \cdot 3 a^{\frac{5}{3}} b^{8} \), we first multiply the coefficients and then combine the powers of \( a \) and \( b \). 1. **Multiply the coefficients**: \( 10 \cdot 3 = 30 \). 2. **Combine the powers of \( a \)**: \[ a^{\frac{7}{3}} \cdot a^{\frac{5}{3}} = a^{\frac{7}{3} + \frac{5}{3}} = a^{\frac{12}{3}} = a^{4}. \] 3. **Combine the powers of \( b \)**: \[ b^{-1} \cdot b^{8} = b^{-1 + 8} = b^{7}. \] Putting it all together, we get: \[ 30 a^{4} b^{7}. \] So, the simplified expression is \( 30 a^{4} b^{7} \).