To solve the expression \(\frac{x+7}{x^2-16}\), we can follow these steps:
1. Identify the components in the expression:
- The numerator is \(x + 7\).
- The denominator is \(x^2 - 16\).
2. Factor the denominator completely:
- \(x^2 - 16\) is a difference of squares, which can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = x\) and \(b = 4\).
- Thus, \(x^2 - 16\) can be factored as \((x - 4)(x + 4)\).
So, the expression now looks like:
[tex]\[
\frac{x+7}{(x-4)(x+4)}
\][/tex]
3. Check for any common factors between the numerator and the denominator:
- The numerator \(x + 7\) and the factors of the denominator \((x - 4)(x + 4)\) have no common factors. Therefore, \(\frac{x + 7}{(x - 4)(x + 4)}\) cannot be simplified further by cancellation.
4. State the final simplified form:
- Since there are no common factors to cancel out, the expression remains as:
[tex]\[
\frac{x + 7}{(x - 4)(x + 4)}
\][/tex]
Thus, the simplified expression for [tex]\(\frac{x+7}{x^2-16}\)[/tex] is [tex]\(\frac{x + 7}{(x - 4)(x + 4)}\)[/tex].
