Question

How do you simplify `\frac { \frac { 1} { a } - \frac { 1} { b } } { \frac { a } { b } - \frac { b } { a } }`?

Answer 1

`(1/a-1/b)/(a/b-b/a) = -1/(a+b)`

Explanation:

Multiply both numerator and denominator by `ab`, then simplify using the difference of squares identity:

`a^2-b^2 = (a-b)(a+b)`

as follows:

`(1/a-1/b)/(a/b-b/a) = (1/a-1/b)/(a/b-b/a)*(ab)/(ab)`

`color(white)((1/a-1/b)/(a/b-b/a)) = (b-a)/(a^2-b^2)`

`color(white)((1/a-1/b)/(a/b-b/a)) = (-color(red)(cancel(color(black)((a-b)))))/(color(red)(cancel(color(black)((a-b))))(a+b))`

`color(white)((1/a-1/b)/(a/b-b/a)) = -1/(a+b)`

(provided `a!=b`)