C. \( \frac{8 a^{3} b}{5 c} \div \frac{10 a b^{2}}{3 c} \)

To simplify the expression: \[ \frac{8 a^{3} b}{5 c} \div \frac{10 a b^{2}}{3 c} \] **Step 1: Rewrite the Division as Multiplication** Dividing by a fraction is the same as multiplying by its reciprocal. So, \[ \frac{8 a^{3} b}{5 c} \div \frac{10 a b^{2}}{3 c} = \frac{8 a^{3} b}{5 c} \times \frac{3 c}{10 a b^{2}} \] **Step 2: Multiply the Numerators and Denominators** \[ \frac{8 a^{3} b \times 3 c}{5 c \times 10 a b^{2}} = \frac{24 a^{3} b c}{50 a b^{2} c} \] **Step 3: Simplify the Constants** \[ \frac{24}{50} = \frac{12}{25} \quad (\text{divided numerator and denominator by 2}) \] **Step 4: Simplify the Variables** - **For \( a \):** \[ \frac{a^{3}}{a} = a^{2} \] - **For \( b \):** \[ \frac{b}{b^{2}} = \frac{1}{b} \] - **For \( c \):** \[ \frac{c}{c} = 1 \] **Step 5: Combine the Simplified Parts** \[ \frac{12}{25} \times a^{2} \times \frac{1}{b} = \frac{12 a^{2}}{25 b} \] **Final Simplified Expression:** \[ \frac{12 a^{2}}{25 b} \]