Question

Write `(x^-1 - y^-1)/(x^-2 - y^-2)` in the simplest form.

Answer 1

Given: `(x^-1 - y^-1)/(x^-2 - y^-2)`

Step-wise calculation:

1. Rewrite the negative powers as fractions:

`(1/x - 1/y)/(1/x^2 - 1/y^2)`

2. Find a common denominator in the numerator:

`((y - x)/(xy))/(1/x^2 - 1/y^2)`

3. Recognize the difference of squares in the denominator: 

`1/x^2 - 1/y^2 = (y^2 - x^2)/(x^2y^2)`

So, `((y - x)/(xy))/((y^2 - x^2)/(x^2y^2)) = (y - x)/(xy) xx (x^2y^2)/(y^2 - x^2)`

4. Simplify the (x) and (y) terms:

`(y - x) xx (xy)/(y^2 - x^2)`

5. Factor the denominator as difference of squares: 

`y^2 - x^2 = (y - x)(y + x)`

6. Substitute back:

`(y - x) xx (xy)/((y - x)(y + x)) = (xy)/(y + x)`

7. Cancel (y – x) terms.

Thus, the simplest form of `(x^-1 - y^-1)/(x^-2 - y^-2)` is `(x y)/(x + y)`.