Simplifying algebraic fractions

An algebraic fraction such as \(\dfrac{12x^3y}{18xy^4}\) has an algebraic expression in its numerator, its denominator or both. It has a value only when its denominator is not zero. As with number fractions, multiplying or dividing the numerator and the denominator by the same non-zero expression gives an equivalent fraction: \(\dfrac{A}{B}=\dfrac{A\times C}{B\times C}\), where \(B\ne0\) and \(C\ne0\).

Why cancelling works: write out the factors

Dividing the numerator and the denominator by their highest common factor \(6xy\) undoes multiplying both by \(6xy\), so the fraction stays equivalent and its value does not change. What is left over is \(\dfrac{2x^2}{3y^3}\).

Method

1. Find the highest common factor (HCF) of the number parts.
2. For each letter, the lower power is common to the numerator and the denominator.
3. Divide the numerator and the denominator by the HCF of the whole expressions.
4. State the restriction: every letter in the original denominator must be non-zero.

Example

Simplify \(\dfrac{15a^3b^2}{25ab^4}\).

\(\begin{aligned}\dfrac{15a^3b^2}{25ab^4}&=\dfrac{5ab^2\times3a^2}{5ab^2\times5b^2} \\ &=\dfrac{3a^2}{5b^2},\qquad a\ne0,\ b\ne0\end{aligned}\)

Check: a letter that cancels completely leaves \(1\), not \(0\). For example, \(\dfrac{4pq}{12pq}=\dfrac13\). A common error is to cancel \(\dfrac{b^2}{b^4}\) to \(b^2\); there are more factors of \(b\) in the denominator, so it is \(\dfrac{1}{b^2}\). Substituting simple values such as \(a=1\), \(b=2\) into both forms (\(\dfrac{60}{400}\) and \(\dfrac{3}{20}\)) is a quick check.