9. \( \frac{9 a^{7} b\left(3 a^{6} b^{2}\right)}{18\left(a^{5} b^{3}\right)} \)

To simplify the expression: \[ \frac{9 a^{7} b \cdot \left(3 a^{6} b^{2}\right)}{18 \cdot \left(a^{5} b^{3}\right)} \] **Step 1: Multiply the Numerator** Multiply the constants and combine the like terms for \( a \) and \( b \): \[ 9 \cdot 3 = 27 \] \[ a^{7} \cdot a^{6} = a^{13} \] \[ b \cdot b^{2} = b^{3} \] So, the numerator becomes: \[ 27 a^{13} b^{3} \] **Step 2: Multiply the Denominator** Multiply the constants and retain the variables: \[ 18 \cdot a^{5} b^{3} = 18 a^{5} b^{3} \] **Step 3: Form the Fraction** \[ \frac{27 a^{13} b^{3}}{18 a^{5} b^{3}} \] **Step 4: Simplify the Coefficients** \[ \frac{27}{18} = \frac{3}{2} \] **Step 5: Subtract the Exponents for Like Bases** \[ a^{13} / a^{5} = a^{8} \] \[ b^{3} / b^{3} = b^{0} = 1 \] **Final Simplified Expression:** \[ \frac{3}{2} a^{8} \] **Alternatively, it can be written as:** \[ \frac{3a^{8}}{2} \]