To find the least common multiple (LCM) of the expressions [tex]\(6 w^2 x^6\)[/tex] and [tex]\(10 w^2 x^3 v^7\)[/tex], we should follow these steps:
1. Identify the coefficients:
- For [tex]\(6 w^2 x^6\)[/tex], the coefficient is [tex]\(6\)[/tex].
- For [tex]\(10 w^2 x^3 v^7\)[/tex], the coefficient is [tex]\(10\)[/tex].
2. Find the LCM of the coefficients:
To find the LCM of [tex]\(6\)[/tex] and [tex]\(10\)[/tex]:
- The prime factorizations are:
[tex]\(6 = 2 \times 3\)[/tex]
[tex]\(10 = 2 \times 5\)[/tex]
- The LCM is found by taking the highest power of each prime that appears in the factorizations:
- The highest power of [tex]\(2\)[/tex] is [tex]\(2\)[/tex].
- The highest power of [tex]\(3\)[/tex] is [tex]\(3\)[/tex].
- The highest power of [tex]\(5\)[/tex] is [tex]\(5\)[/tex].
- Therefore, the LCM of the coefficients is [tex]\(2 \times 3 \times 5 = 30\)[/tex].
3. Determine the LCM for each variable:
- For [tex]\(w\)[/tex], the highest power in both expressions is [tex]\(w^2\)[/tex].
- For [tex]\(x\)[/tex], the highest power is [tex]\(x^6\)[/tex] (as [tex]\(6 > 3\)[/tex]).
- For [tex]\(v\)[/tex], it appears only in the second term as [tex]\(v^7\)[/tex], so its maximum power is [tex]\(v^7\)[/tex].
4. Combine the results:
We take the LCM of the coefficients and the highest powers of each variable involved to form the combined LCM expression.
Thus, combining all these factors:
- The LCM of the coefficients is [tex]\(30\)[/tex].
- The LCM of the variables and their highest exponents are [tex]\(w^2\)[/tex], [tex]\(x^6\)[/tex], and [tex]\(v^7\)[/tex].
Therefore, the least common multiple of the two expressions [tex]\(6 w^2 x^6\)[/tex] and [tex]\(10 w^2 x^3 v^7\)[/tex] is:
[tex]\[30 w^2 x^6 v^7\][/tex]
