To find the least common multiple (LCM) of the algebraic expressions [tex]\(6u^8 v^4 w^5 \text{ and } 10 v^3 w^6\)[/tex], we need to follow these steps:
1. Determine the LCM of the coefficients:
The coefficients of the expressions are 6 and 10.
- The prime factorization of 6 is [tex]\(2 \times 3\)[/tex].
- The prime factorization of 10 is [tex]\(2 \times 5\)[/tex].
To find the LCM, we take the highest power of each prime number that appears in the factorizations:
- The highest power of 2 is [tex]\(2^1\)[/tex].
- The highest power of 3 is [tex]\(3^1\)[/tex].
- The highest power of 5 is [tex]\(5^1\)[/tex].
Thus, the LCM of 6 and 10 is [tex]\(2^1 \times 3^1 \times 5^1 = 30\)[/tex].
2. Determine the LCM of each variable:
For each variable [tex]\(u\)[/tex], [tex]\(v\)[/tex], and [tex]\(w\)[/tex], we look at the highest power that appears in the expressions.
- Variable [tex]\(u\)[/tex]:
- In the first expression, [tex]\(u\)[/tex] has a power of 8.
- In the second expression, [tex]\(u\)[/tex] does not appear, which means it has a power of 0.
The highest power of [tex]\(u\)[/tex] is [tex]\(u^8\)[/tex].
- Variable [tex]\(v\)[/tex]:
- In the first expression, [tex]\(v\)[/tex] has a power of 4.
- In the second expression, [tex]\(v\)[/tex] has a power of 3.
The highest power of [tex]\(v\)[/tex] is [tex]\(v^4\)[/tex].
- Variable [tex]\(w\)[/tex]:
- In the first expression, [tex]\(w\)[/tex] has a power of 5.
- In the second expression, [tex]\(w\)[/tex] has a power of 6.
The highest power of [tex]\(w\)[/tex] is [tex]\(w^6\)[/tex].
3. Combine all parts to obtain the LCM of the given expressions:
Therefore, the LCM of the expressions [tex]\(6 u^8 v^4 w^5 \text{ and } 10 v^3 w^6\)[/tex] is obtained by combining the LCM of the coefficients and the highest powers of each variable:
[tex]\[
\text{LCM} = 30 \cdot u^8 \cdot v^4 \cdot w^6
\][/tex]
Summarizing, the least common multiple of the expressions [tex]\(6u^8v^4w^5\)[/tex] and [tex]\(10v^3w^6\)[/tex] is:
[tex]\[
30 u^8 v^4 w^6
\][/tex]
