Question

select the correct answer.
assuming no denominator equals zero, which expression is equivalent to the given expression?
image of the fraction: numerator is 1 over (a² - 2a - 8), denominator is (a - 5)/(a - 4)
options:

1. (a - 4)/(a + 2)
2. 1/(a + 2)(a - 5)
3. (a + 2)/(a - 5)
4. (a - 5)/(a - 4)²

Explanation:

Step1: Rewrite complex fraction as multiplication

Dividing by a fraction is multiplying by its reciprocal:
$$\frac{1}{a^2-2a-8} \div \frac{a-5}{a-4} = \frac{1}{a^2-2a-8} \times \frac{a-4}{a-5}$$

Step2: Factor the quadratic denominator

Factor $a^2-2a-8$ into binomials:
$$a^2-2a-8 = (a-4)(a+2)$$
Substitute back:
$$\frac{1}{(a-4)(a+2)} \times \frac{a-4}{a-5}$$

Step3: Cancel common terms

Cancel $(a-4)$ from numerator and denominator:
$$\frac{1}{(a+2)} \times \frac{1}{a-5} = \frac{1}{(a+2)(a-5)}$$

Answer:

$\frac{1}{(a+2)(a-5)}$ (second option)