C2.3; E2.3 - Simplifying algebraic fractions | Cambridge IGCSE International Mathematics

Is $\frac{6x}{9x}$ equal to $\frac23$ for every value of $x$ ? Cancelling common factors makes an algebraic fraction shorter, but it does not remove the original denominator's restrictions. This lesson develops one-step simplification in C2.3 and E2.3, including powers and the difference between factors and terms.

Cancelling factors

A numerical fraction keeps its value when you divide its numerator and denominator by the same non-zero factor: $\frac69=\frac23$ . Letters can represent factors too. If $x\ne0$ , the numerator and denominator of $\frac{6x}{9x}$ share the factor $3x$ .

Algebraic fraction

An algebraic fraction has algebraic expressions in its numerator or denominator. Its denominator must not be zero. Cancelling divides the complete numerator and complete denominator by a common non-zero factor.

Think of $\frac{3\times2\times x}{3\times3\times x}$ : removing matching non-zero factors leaves $\frac23$ . The denominator condition is part of the equality, even when the simplified result no longer contains a denominator with a letter.

Worked example 1: cancelling a numerical factor

Question. Simplify $\frac{8x}{12}$ .

Plan. Divide the numerator and denominator by their common numerical factor 4.

\begin{aligned} \frac{8x}{12}&=\frac{4\times2x}{4\times3}\\ &=\frac{2x}{3} &&\text{cancel the factor 4} \end{aligned}

Answer. $\frac{2x}{3}$ .

Check. At $x=3$ , the original fraction is $\frac{24}{12}=2$ ; the simplified form is $\frac63=2$ . Multiplying its top and bottom by 4 also recovers the original fraction for any $x$ .

Move the letter to the denominator. The common numerical factor still cancels, but the remaining letter stays below the fraction line.

Worked example 2: a letter in the denominator

Question. Simplify $\frac8{12x}$ , where $x\ne0$ .

Plan. Cancel 4, keeping the remaining factor $x$ in the denominator.

\begin{aligned} \frac8{12x}&=\frac{4\times2}{4\times3x}\\ &=\frac2{3x} \end{aligned}

Answer. $\frac2{3x}$ , for $x\ne0$ .

Check. At $x=2$ , both forms give $\frac13$ . Multiplying top and bottom of $\frac2{3x}$ by 4 recovers $\frac8{12x}$ .

For the opening fraction, cancellation of $3x$ gives $\frac{6x}{9x}=\frac23$ , provided $x\ne0$ . Cancelling does not mean deleting every letter you see: a factor needs a matching factor on the other side of the fraction line.

Watch out

$\frac3{6x}=\frac1{2x}$ , not $\frac x2$ . The position of the uncancelled factor is unchanged. Substitution with $x=2$ exposes the wrong answer: $\frac14\ne1$ .

Cancel only a factor that appears in both complete products.

Powers in fractions

An index counts repeated factors. The fraction $\frac{x^3}{x}$ contains three factors of $x$ above the line and one below. If $x\ne0$ , cancelling one pair leaves two factors above, giving $x^2$ .

You can use this factor-counting explanation without learning a new rule. When the denominator has more copies, the uncancelled factors stay in the denominator. If every variable factor cancels, the remaining factor is 1, not 0.

Worked example 3: factors left above the line

Question. Simplify $\frac{p^4}{p^2}$ , where $p\ne0$ .

Plan. Cancel the two denominator factors with two matching numerator factors.

\begin{aligned} \frac{p^4}{p^2}&=\frac{p\times p\times p\times p}{p\times p}\\ &=p\times p=p^2 \end{aligned}

Answer. $p^2$ , for $p\ne0$ .

Check. Multiplying $p^2$ by the original denominator $p^2$ gives $p^4$ , the original numerator. At $p=2$ , $16\div4=4$ .

Reverse which side contains the larger power. The cancellation is the same, but the surviving factors are now below the line.

Worked example 4: factors left below the line

Question. Simplify $\frac{p^2}{p^4}$ , where $p\ne0$ .

Plan. Cancel the two numerator factors, leaving two denominator factors.

\begin{aligned} \frac{p^2}{p^4}&=\frac{p\times p}{p\times p\times p\times p}\\ &=\frac1{p\times p}=\frac1{p^2} \end{aligned}

Answer. $\frac1{p^2}$ , for $p\ne0$ .

Check. $\frac1{p^2}\times p^4=p^2$ , the original numerator. At $p=2$ , $4\div16=\frac14$ , agreeing with $\frac1{2^2}$ .

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C2.3; E2.3 - Simplifying algebraic fractions

Cancelling factors

Powers in fractions

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