To solve the given expression [tex]\(\frac{a^2 - 6a}{a - 6} \cdot \frac{a + 3}{a}\)[/tex], let's perform the indicated operations step-by-step and then simplify the result.
1. Analyze the first fraction [tex]\(\frac{a^2 - 6a}{a - 6}\)[/tex]:
- The numerator is [tex]\(a^2 - 6a\)[/tex]. This expression can be factored by taking out the common factor [tex]\(a\)[/tex]:
[tex]\[
a^2 - 6a = a(a - 6)
\][/tex]
- So, [tex]\(\frac{a^2 - 6a}{a - 6}\)[/tex] converts to:
[tex]\[
\frac{a(a - 6)}{a - 6}
\][/tex]
2. Simplify the first fraction:
- Note that [tex]\(\frac{a(a - 6)}{a - 6}\)[/tex] simplifies to [tex]\(a\)[/tex] if [tex]\(a \neq 6\)[/tex], because the [tex]\(a - 6\)[/tex] terms cancel each other out:
[tex]\[
\frac{a(a - 6)}{a - 6} = a
\][/tex]
- Thus, the first fraction simplifies to [tex]\(a\)[/tex].
3. Combine the simplified first fraction with the second fraction [tex]\(\frac{a + 3}{a}\)[/tex]:
- Now we have:
[tex]\[
a \cdot \frac{a + 3}{a}
\][/tex]
- This multiplication can be performed by recognizing that [tex]\(a\)[/tex] in the numerator and [tex]\(a\)[/tex] in the denominator cancel out:
[tex]\[
a \cdot \frac{a + 3}{a} = a + 3
\][/tex]
Hence, the simplified result of the given operation is:
[tex]\[
a + 3
\][/tex]
Therefore, the correct answer is:
[tex]\[
a + 3
\][/tex]
