## Precalculus (6th Edition) Blitzer

$\displaystyle \frac{1}{xy},\qquad x\neq 0, y\neq 0,x\neq-y$
Complex rational expressions have rational expressions in the numerator or/and in the denominator. Here, the numerator contains $\displaystyle \frac{1}{x}$ and $\displaystyle \frac{1}{y}$ whose LCD =$xy.$ (Exclusions from the domain are $x\neq 0, y\neq 0.)$ To get rid of these fractions, we multiply both the numerator and denominator with $xy.$ $\displaystyle \frac{xy}{xy}\times\frac{\frac{1}{x}+\frac{1}{y}}{x+y}=\frac{y+x}{xy(x+y)},\qquad x\neq 0, y\neq 0,x\neq-y$ The expression has a common factor. Reduce. $=\displaystyle \frac{1}{xy},\qquad x\neq 0, y\neq 0,x\neq-y$