Question

perform the following operation and express in simplest form.
\\(\frac{x + 8}{x^2 - 64} div \frac{x - 4}{x^2 + x - 72}\\)

Explanation:

Step1: Factor the denominators

First, factor \(x^{2}-64\) as a difference of squares: \(x^{2}-64=(x + 8)(x - 8)\).
Then, factor \(x^{2}+x - 72\). We need two numbers that multiply to \(-72\) and add to \(1\). Those numbers are \(9\) and \(-8\), so \(x^{2}+x - 72=(x + 9)(x - 8)\).

The expression becomes: \(\frac{x + 8}{(x + 8)(x - 8)}\div\frac{x - 4}{(x + 9)(x - 8)}\)

Step2: Rewrite division as multiplication

Dividing by a fraction is the same as multiplying by its reciprocal. So we rewrite the division as:
\(\frac{x + 8}{(x + 8)(x - 8)}\times\frac{(x + 9)(x - 8)}{x - 4}\)

Step3: Cancel common factors

Cancel out the common factors \((x + 8)\) and \((x - 8)\) from the numerator and denominator:
After canceling, we have \(\frac{1}{1}\times\frac{(x + 9)}{x - 4}=\frac{x + 9}{x - 4}\)

Answer:

\(\frac{x + 9}{x - 4}\)