To find the least common multiple (LCM) of the expressions [tex]\(a^2\)[/tex] and [tex]\(a^3\)[/tex], we need to determine the smallest expression that both [tex]\(a^2\)[/tex] and [tex]\(a^3\)[/tex] will divide without leaving a remainder.
Step-by-step solution:
1. Identify the Expressions:
We are given two expressions:
- [tex]\(a^2\)[/tex]
- [tex]\(a^3\)[/tex]
2. Understand the LCM Concept:
The LCM of two algebraic expressions is the smallest expression that is divisible by both of the given expressions.
3. Prime Factorization (if necessary):
For polynomial expressions like these, we look at the powers of the variable [tex]\(a\)[/tex]. Here:
- [tex]\(a^2\)[/tex] is [tex]\(a\)[/tex] raised to the power of 2.
- [tex]\(a^3\)[/tex] is [tex]\(a\)[/tex] raised to the power of 3.
4. Choose the Highest Power:
To find the LCM, we take the highest power of [tex]\(a\)[/tex] that appears in the factorizations of the given expressions. In this case:
- The highest power of [tex]\(a\)[/tex] in [tex]\(a^2\)[/tex] is [tex]\(2\)[/tex].
- The highest power of [tex]\(a\)[/tex] in [tex]\(a^3\)[/tex] is [tex]\(3\)[/tex].
Therefore, the highest power of [tex]\(a\)[/tex] appearing in either expression is [tex]\(3\)[/tex].
5. State the LCM:
Given that [tex]\(a^3\)[/tex] is the expression with the highest power of [tex]\(a\)[/tex] present in both [tex]\(a^2\)[/tex] and [tex]\(a^3\)[/tex], the LCM of [tex]\(a^2\)[/tex] and [tex]\(a^3\)[/tex] is simply [tex]\(a^3\)[/tex].
Therefore, the LCM of [tex]\(a^2\)[/tex] and [tex]\(a^3\)[/tex] is:
[tex]\[
\boxed{a^3}
\][/tex]
