Question

How do you simplify `(\frac { a ^ { - 1} + b ^ { - 1} } { a ^ { - 2} - b ^ { - 2} } ) ^ { 2}`?

Answer 1

This simpifies all the way to `(ab)/(b-a)`
Details follow...

Explanation:

A negative exponent can be changed to a positive exponent by moving it to the denominator of a fraction.

That is, `a^(-1)` is equal to `1/(a^1)` (or simply `1/a`) and `a^(-2)` is equal to `1/a^2`

If we apply this to every term in the expression, it becomes

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

Now, we must add (or subtract) the fractions. This requires a common denominator for both the top expression and the bottom:
This common denominator is found by multiplying each fraction by the denominator in the other fraction.

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

Combining:

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

The `ab` in the top fraction cancels `ab` from the bottom fraction:

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

Next, we can factor `b^2-a^2` into `(b+a)(b-a)`

`((b+a)/(((b+a)(b-a))/(ab)))`

One more cancellation, of `(b+a)`:

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

Finally, invert the denominator:

`(ab)/(b-a)`