Fractions. Simplifying Fractions

Fractions are in simplest form when the numerator and denominator have no common factor other than 1. To simplify a fraction, factor both the numerator and denominator.
Don't cancel terms that are not common factors. Below is a common mistake:

$$\frac{x^{2}-4}{8}\neq \frac{x^{2}}{2}$$

$x^{2}-4$ and 8 do not have a common factor, so this expression is already in simplest form.
EXAMPLE:
Simplify $\displaystyle \frac{(3x+12)}{(3x+3y)}$
$\displaystyle =\frac{3(x+4)}{3(x+y)}$    Factor the numerator and denominator. 3 is a common factor.
$\displaystyle =\frac{(x+4)}{(x+y)},\; x\neq -y$  (This restriction is important because you cannot (x+y) divide by zero!) (Answer)