Find the least common multiple. \( 5 j^{4}, 6 j^{5} \) Write your answer as a constant times a product of single variables raised to exponents.

To find the least common multiple (LCM) of the expressions \( 5 j^{4} \) and \( 6 j^{5} \), we will follow these steps: 1. **Identify the coefficients**: The coefficients are \( 5 \) and \( 6 \). 2. **Find the LCM of the coefficients**: The LCM of \( 5 \) and \( 6 \) can be calculated by finding the smallest multiple that both numbers share. 3. **Identify the variable parts**: The variable parts are \( j^{4} \) and \( j^{5} \). 4. **Find the LCM of the variable parts**: The LCM of \( j^{4} \) and \( j^{5} \) is determined by taking the variable with the highest exponent. Now, let's calculate the LCM step by step. ### Step 1: LCM of the coefficients The prime factorization of the coefficients: - \( 5 = 5^1 \) - \( 6 = 2^1 \times 3^1 \) The LCM is found by taking the highest power of each prime: - For \( 2 \): \( 2^1 \) - For \( 3 \): \( 3^1 \) - For \( 5 \): \( 5^1 \) Thus, the LCM of the coefficients is: \[ \text{LCM}(5, 6) = 2^1 \times 3^1 \times 5^1 = 30 \] ### Step 2: LCM of the variable parts The variable parts are \( j^{4} \) and \( j^{5} \). The LCM is: \[ \text{LCM}(j^{4}, j^{5}) = j^{\max(4, 5)} = j^{5} \] ### Final LCM Combining both results, we have: \[ \text{LCM}(5 j^{4}, 6 j^{5}) = 30 j^{5} \] Thus, the least common multiple is: \[ \boxed{30 j^{5}} \]