Question

For all values for which the fraction is defined, how do you simplify `(a^-1-b^-1)/(b^-2-a^-2)`?

Answer 1

`(a^(-1)-b^(-1))/(b^(-2)-a^(-2)) = -(ab)/(a+b)`

Explanation:

Multiply both numerator and denominator by `a^2b^2` and simplify as follows:

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

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

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

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

`color(white)((a^(-1)-b^(-1))/(b^(-2)-a^(-2))) = -(abcolor(red)(cancel(color(black)((a-b)))))/(color(red)(cancel(color(black)((a-b))))(a+b))`

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