Find the least common multiple of these two expressions. \( 6 y^{3} x^{2} \) and \( 9 y^{5} x^{6} v \)

To find the least common multiple (LCM) of the expressions \( 6 y^{3} x^{2} \) and \( 9 y^{5} x^{6} v \), we can follow these steps: 1. **Factor each expression into its prime factors.** - For \( 6 y^{3} x^{2} \): - The number \( 6 \) can be factored as \( 2 \times 3 \). - Therefore, \( 6 y^{3} x^{2} = 2 \times 3 \times y^{3} \times x^{2} \). - For \( 9 y^{5} x^{6} v \): - The number \( 9 \) can be factored as \( 3^2 \). - Therefore, \( 9 y^{5} x^{6} v = 3^2 \times y^{5} \times x^{6} \times v \). 2. **Identify the highest power of each factor present in both expressions.** - For the factor \( 2 \): - It appears in \( 6 y^{3} x^{2} \) as \( 2^1 \) and does not appear in the second expression. - Highest power: \( 2^1 \). - For the factor \( 3 \): - It appears as \( 3^1 \) in the first expression and \( 3^2 \) in the second expression. - Highest power: \( 3^2 \). - For the factor \( y \): - It appears as \( y^3 \) in the first expression and \( y^5 \) in the second expression. - Highest power: \( y^5 \). - For the factor \( x \): - It appears as \( x^2 \) in the first expression and \( x^6 \) in the second expression. - Highest power: \( x^6 \). - For the factor \( v \): - It does not appear in the first expression and appears as \( v^1 \) in the second expression. - Highest power: \( v^1 \). 3. **Combine these highest powers to form the LCM.** Therefore, the LCM is: \[ LCM = 2^1 \times 3^2 \times y^5 \times x^6 \times v^1 \] 4. **Write the final expression:** \[ LCM = 2 \times 9 \times y^5 \times x^6 \times v = 18 y^5 x^6 v \] Thus, the least common multiple of \( 6 y^{3} x^{2} \) and \( 9 y^{5} x^{6} v \) is \( 18 y^5 x^6 v \).