Question

perform the following operation and express in simplest form.\\(\dfrac{7x}{x - 8} \div \dfrac{x^2 + 4x - 32}{x^2 - 64}\\)

Explanation:

Step1: Rewrite division as multiplication

$\frac{7x}{x-8} \times \frac{x^2 - 64}{x^2 + 4x - 32}$

Step2: Factor all polynomials

$x^2 - 64=(x-8)(x+8)$; $x^2 + 4x - 32=(x+8)(x-4)$
Substitute factored forms:
$\frac{7x}{x-8} \times \frac{(x-8)(x+8)}{(x+8)(x-4)}$

Step3: Cancel common factors

Cancel $(x-8)$ and $(x+8)$ from numerator and denominator:
$\frac{7x}{1} \times \frac{1}{x-4}$

Step4: Multiply remaining terms

$\frac{7x}{x-4}$

Answer:

$\frac{7x}{x-4}$ (where $x
eq 8, -8, 4$)