How would i begin factoring this expression:
(x^-2)-(y^-2)/(x^-1)-(y^-1)
(reads x to the negative 2 minus y to the negative 2...)
I just need somewhere to start...
Try writing them with positive powers first...
$\displaystyle \frac{x^{-2}-y^{-2}}{x^{-1} - y^{-1}} = \frac{\frac{1}{x^2} - \frac{1}{y^2}}{\frac{1}{x} - \frac{1}{y}}$
$\displaystyle = \frac{\frac{y^2 - x^2}{x^2y^2}}{\frac{y - x}{xy}}$
$\displaystyle = \frac{xy(y^2 - x^2)}{x^2y^2(y - x)}$
$\displaystyle = \frac{y^2 - x^2}{xy(y - x)}$
$\displaystyle = \frac{(y - x)(y + x)}{xy(y - x)}$
$\displaystyle = \frac{y + x}{xy}$.
The first thing I would do is multiply both numerator and denominator by $\displaystyle x^2y^2$:
$\displaystyle \frac{(x^{-2}-y^{-2})(x^2y^2)}{(x^{-1}- y^{-1})(x^2y^2)}= \frac{y^2- x^2}{xy^2- x^2y}$
You should be able to factor $\displaystyle y^2- x^2$ easily and $\displaystyle xy^2- x^2y= xy(y- x)$.