Find the common factors of the given terms in each 1) \( 12 x^{2} y, 18 x y^{2} \)

To find the **greatest common factor (GCF)** of the terms \( 12x^{2}y \) and \( 18xy^{2} \), follow these steps: 1. **Factor Each Term:** - **For \( 12x^{2}y \):** \[ 12x^{2}y = 2^{2} \times 3 \times x^{2} \times y \] - **For \( 18xy^{2} \):** \[ 18xy^{2} = 2 \times 3^{2} \times x \times y^{2} \] 2. **Identify the Common Factors:** - **Numerical Factors:** - Both terms have at least one \( 2 \) and one \( 3 \). - Take the **minimum** exponent for each common factor: \[ 2^{1} \times 3^{1} = 6 \] - **Variable Factors:** - For \( x \): - Minimum exponent is \( 1 \) (from \( x \) and \( x^{2} \)). - For \( y \): - Minimum exponent is \( 1 \) (from \( y \) and \( y^{2} \)). - Combine the common numerical and variable factors: \[ 6 \times x \times y = 6xy \] 3. **Conclusion:** The **greatest common factor** of \( 12x^{2}y \) and \( 18xy^{2} \) is: \[ \boxed{6xy} \]