Find the least common multiple of these two expressions.12v3u7  and  18x6v2u8

To find the least common multiple (LCM) of the two expressions 12v^3u^7 and 18x^6v^2u^8, we need to break down each expression into its prime factors and then combine them in a way that includes each factor the greatest number of times it appears in either expression.

Step 1: Prime factorization of the coefficients

• The prime factorization of 12 is 2^2 * 3
• The prime factorization of 18 is 2 * 3^2

Step 2: Identify the highest power of each variable in the expressions

• For v, the highest power is 3 (from the first expression)
• For u, the highest power is 8 (from the second expression)
• For x, the highest power is 6 (from the second expression)

Step 3: Combine the prime factors and variables with the highest powers

• For the prime factors, take the highest power of each factor: 2^2 and 3^2
• For the variables, take the highest power of each variable: v^3, u^8, and x^6

Step 4: Multiply these together to get the LCM

• LCM = 2^2 * 3^2 * v^3 * u^8 * x^6
• LCM = 4 * 9 * v^3 * u^8 * x^6
• LCM = 36v^3u^8x^6

So, the least common multiple of 12v^3u^7 and 18x^6v^2u^8 is 36v^3u^8x^6.