To simplify the given expression:
[tex]\[
\frac{7}{a+8} + \frac{7}{a^2 - 64}
\][/tex]
Let's go through the steps:
1. Factor the Denominator:
We notice that [tex]\(a^2 - 64\)[/tex] is a difference of squares, which can be factored as:
[tex]\[
a^2 - 64 = (a - 8)(a + 8)
\][/tex]
Therefore, the second fraction becomes:
[tex]\[
\frac{7}{(a - 8)(a + 8)}
\][/tex]
2. Combine with a Common Denominator:
The first fraction has [tex]\(a + 8\)[/tex] in the denominator. To combine these two fractions, they should have a common denominator, which is [tex]\((a - 8)(a + 8)\)[/tex]:
[tex]\[
\frac{7}{a + 8} = \frac{7(a - 8)}{(a + 8)(a - 8)}
\][/tex]
Now, we can write the original expression as:
[tex]\[
\frac{7(a - 8)}{(a + 8)(a - 8)} + \frac{7}{(a + 8)(a - 8)}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
\frac{7(a - 8) + 7}{(a + 8)(a - 8)} = \frac{7(a - 8) + 7}{a^2 - 64}
\][/tex]
3. Simplify the Numerator:
Expand and simplify the numerator:
[tex]\[
7(a - 8) + 7 = 7a - 56 + 7 = 7a - 49
\][/tex]
So the expression becomes:
[tex]\[
\frac{7a - 49}{a^2 - 64}
\][/tex]
4. Check Possible Equivalent Expressions:
Looking at the provided answer options:
[tex]\[
\frac{7}{a+8} + \frac{7}{a^2-64}
\][/tex]
We can verify the simplified form is:
[tex]\[
\frac{7a - 49}{(a-8)(a+8)}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\frac{7a - 49}{(a - 8)(a + 8)}
\][/tex]
