Sure, let's simplify the given mathematical expression step by step:
We start with the expression:
[tex]\[
\frac{a^2 + 2ab}{6a}
\][/tex]
### Step-by-Step Simplification:
1. Identify Terms in the Numerator:
The numerator of our fraction is [tex]\(a^2 + 2ab\)[/tex].
2. Factor Out the Common Factor in the Numerator:
Notice that in the numerator [tex]\(a^2 + 2ab\)[/tex], there is a common factor of [tex]\(a\)[/tex] in both terms.
We can factor [tex]\(a\)[/tex] out:
[tex]\[
a^2 + 2ab = a(a + 2b)
\][/tex]
Thus, our expression now looks like this:
[tex]\[
\frac{a(a + 2b)}{6a}
\][/tex]
3. Cancel Out the Common Factor in the Numerator and Denominator:
We can see that there is an [tex]\(a\)[/tex] in both the numerator and the denominator:
[tex]\[
\frac{a(a + 2b)}{6a}
\][/tex]
Dividing both the numerator and the denominator by [tex]\(a\)[/tex] (given that [tex]\(a \neq 0\)[/tex]):
[tex]\[
\frac{a(a + 2b)}{6a} = \frac{a + 2b}{6}
\][/tex]
4. Separate the Terms in the Fraction:
Now, we can write the simplified fraction by splitting it into two separate terms:
[tex]\[
\frac{a + 2b}{6} = \frac{a}{6} + \frac{2b}{6}
\][/tex]
5. Simplify the Individual Terms:
Simplify each term in the expression:
[tex]\[
\frac{a}{6} + \frac{2b}{6} = \frac{a}{6} + \frac{b}{3}
\][/tex]
### Final Simplified Expression:
The simplified version of the given expression [tex]\(\frac{a^2 + 2ab}{6a}\)[/tex] is:
[tex]\[
\frac{a}{6} + \frac{b}{3}
\][/tex]
So, the original expression [tex]\(\frac{a^2 + 2ab}{6a}\)[/tex] simplifies to:
[tex]\[
\frac{a}{6} + \frac{b}{3}
\][/tex]
