Questions: Find the least common multiple of these two expressions: (16 w^3 u^7) and (6 v^8 w^5 u^7)

Solution Steps

First, identify the variables and their respective exponents in each monomial:

• For \(16 w^{3} u^{7}\):

  • \(w\) has an exponent of 3
  • \(u\) has an exponent of 7

• For \(6 v^{8} w^{5} u^{7}\):

  • \(v\) has an exponent of 8
  • \(w\) has an exponent of 5
  • \(u\) has an exponent of 7

The coefficients of the monomials are 16 and 6. To find the least common multiple (LCM) of these coefficients:

• Prime factorization of 16: \(2^4\)
• Prime factorization of 6: \(2^1 \times 3^1\)

The LCM is found by taking the highest power of each prime number present in the factorizations:

• \(2^4\) (from 16)
• \(3^1\) (from 6)

Thus, the LCM of the coefficients is \(2^4 \times 3^1 = 48\).

For each variable, take the highest exponent present in the monomials:

• For \(w\), the highest exponent is 5 (from \(w^5\))
• For \(u\), the highest exponent is 7 (from both \(u^7\))
• For \(v\), the highest exponent is 8 (from \(v^8\))

Final Answer