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Perform the indicated operation and simplify the result.

[tex]\[
\frac{3a^2 - 13a + 4}{9a^2 - 6a + 1} \cdot \frac{28 + 7a}{a^2 - 16}
\][/tex]


Sagot :

Certainly! Let's perform the indicated operation and simplify the result step-by-step.

We start with the given fractions:
[tex]\[ \frac{3 a^2 - 13 a + 4}{9 a^2 - 6 a + 1} \cdot \frac{28 + 7 a}{a^2 - 16} \][/tex]

1. Factor the numerator and denominator of both fractions:

For the first fraction:
- Numerator: [tex]\(3 a^2 - 13 a + 4\)[/tex]
- Denominator: [tex]\(9 a^2 - 6 a + 1\)[/tex]

The factors are:
[tex]\[ 3 a^2 - 13 a + 4 = (a - 4)(3 a - 1) \][/tex]
and
[tex]\[ 9 a^2 - 6 a + 1 = (3 a - 1)^2 \][/tex]

For the second fraction:
- Numerator: [tex]\(28 + 7 a\)[/tex]
- Denominator: [tex]\(a^2 - 16\)[/tex]

The factors are:
[tex]\[ 28 + 7 a = 7(a + 4) \][/tex]
and
[tex]\[ a^2 - 16 = (a - 4)(a + 4) \][/tex]

2. Substitute the factored forms into the original expression:

[tex]\[ \frac{(a - 4)(3 a - 1)}{(3 a - 1)^2} \cdot \frac{7(a + 4)}{(a - 4)(a + 4)} \][/tex]

3. Combine the fractions and simplify:

[tex]\[ \frac{(a - 4)(3 a - 1) \cdot 7(a + 4)}{(3 a - 1)^2 \cdot (a - 4)(a + 4)} \][/tex]

Notice that some terms in the numerator and denominator can be cancelled out:

- [tex]\((a - 4)\)[/tex] in both the numerator and the denominator
- [tex]\((a + 4)\)[/tex] in both the numerator and the denominator

After cancelling these terms, we get:

[tex]\[ \frac{7 (3 a - 1)}{(3 a - 1)^2} \][/tex]

4. Simplify the remaining fraction by cancelling out [tex]\((3 a - 1)\)[/tex]:

[tex]\[ \frac{7}{3 a - 1} \][/tex]

Thus, the final simplified result of the operation is:
[tex]\[ \boxed{\frac{7}{3 a - 1}} \][/tex]