Any tips on this problem involving the simplification of a rational function?

[JacobD94]

'm taking pre-calculus, and considering it's been two years since I've done some algebra of this sort, especially with negative exponents within a rational equation. I'll need some enlightenment on what I did wrong in this problem attached in an image, along with my work.

 

[phoenix9124]

'm taking pre-calculus, and considering it's been two years since I've done some algebra of this sort, especially with negative exponents within a rational equation. I'll need some enlightenment on what I did wrong in this problem attached in an image, along with my work.

View attachment 2490View attachment 2491

Quickly observing the problem, i'd do the following:

Multiplying everything by x^2:
(x+6)/(x^2 - 36)

Then I see that the (x^2 - 36) can be rewritten as (x+6)(x-6).

This should help you out.

 

[HallsofIvy]

However, it is important to note that \(\displaystyle \frac{x+6}{(x- 6)(x+ 6)}\) is NOT equal to \(\displaystyle \frac{1}{x- 6}\). They are the same everywhere except at x= -6. The second form is defined at x= -6 (it has value -1/36) but the first is not.