# Simplify equation with negative exponents

1. Aug 23, 2013

### Boba-Feet

1. The problem statement, all variables and given/known data
Simplify (x-2 - y-2) / (x-1 + y-1)

2. Relevant equations

3. The attempt at a solution
So I just factorised the numerator into x-1 - y-1 and x-1 + y-1. And was left with x-1 - y-1 as an answer. The text book gives (y - x) / (xy) as the answer (no working shown). So after some substitution I've realised my answer and the text book's are the same. But I would like to know how they got theirs and is it a simpler form of my answer?

And I read on the internet that polynomials with negative exponents are not called polynomials, is that true and what should they be called? Thanks a lot in advance :)

2. Aug 23, 2013

To simplifying this equation is like simplifying any other fraction with a plus or minus in between it.

evaluate this:
$\frac{1}{10} - \frac{1}{5}$ = ?

Then, imagine that x = 10 and y = 5

3. Aug 23, 2013

### Staff: Mentor

Starting with your result, you can write $x^{-1}=\frac{1}{x}$ and do the same with y, and combine the sum to a single fraction afterwards.

Well, they are polynomials in a different variable ($x^{-1}$ instead of x)...
As functions, they are called rational functions.

4. Aug 23, 2013

### HallsofIvy

MY first thought was to get rid of those negative exponents by multiplying both numerator and denominator by $x^2y^2$. That gives
$$\frac{y^2- x^2}{xy^2- x^2y}= \frac{(y- x)(y+x)}{xy(x- y)}= \frac{y+ x}{xy}$$

5. Aug 23, 2013

### Boba-Feet

OK I am able to see how the text book arrived at it's answer.
I looked at it and initially thought to factorise because I saw the difference of two squares (x^-2 - y^-2).
But the text book writer thought to remove the negative exponents first. Strange that both answers look so different but mean the same thing.

Thanks for the quick and helpful responses.

Last edited: Aug 23, 2013
6. Aug 23, 2013

### Ray Vickson

It is really not strange at all; do you prefer to write (1/3) + (1/5) or 8/15? Which way is "better" depends on what you want to do with the answer.