Find the least common multiple. \( 4 r^{6}, 14 r^{9} \) Write your answer as a constant times a product of single variables raised to exponents.

To find the least common multiple (LCM) of \( 4r^6 \) and \( 14r^9 \), we first determine the LCM of the coefficients and the LCM of the variable terms separately. The coefficients are 4 and 14. The prime factorization of 4 is \( 2^2 \) and for 14 is \( 2^1 \times 7^1 \). The LCM takes the highest power of each prime factor: - For 2, the highest power is \( 2^2 \). - For 7, the highest power is \( 7^1 \). Thus, the LCM of the coefficients is \( 2^2 \times 7^1 = 4 \times 7 = 28 \). For the variable \( r \), we take the highest exponent from the two terms: - \( r^6 \) and \( r^9 \) gives us \( r^9 \). Combining these results, the LCM of \( 4r^6 \) and \( 14r^9 \) is: \[ 28r^9 \] And there you have it—the least common multiple is \( 28r^9 \)!