Multiplying and dividing algebraic fractions

A fraction is defined only when its denominator is nonzero. Cancel common factors of a product, never individual terms of a sum. For division, multiply by the reciprocal; the divisor itself must also be nonzero.

Example

\(\dfrac{3a^2}{4b}\times\dfrac{8b}{9a}=\dfrac{2a}{3}\), with \(a\ne0\) and \(b\ne0\). These restrictions remain even though the simplified expression has no letter in its denominator.

\(\begin{aligned}\dfrac{3a^2}{4}\div\dfrac{9a}{10}&=\dfrac{3a^2}{4}\times\dfrac{10}{9a}\\&=\dfrac{5a}{6},\qquad a\ne0.\end{aligned}\)

For \(x\ne0\), \(\dfrac{x+2}{x}\) does not simplify to \(2\): at \(x=1\), its value is \(3\). The numerator is a sum, so there is no factor \(x\) common to the whole numerator.