Multiplying and dividing rational expressions - Higher

The method to multiply fractions is to multiply the numerators together, multiply the denominators together and then cancel down if necessary.

The method to divide fractions is to keep the first fraction the same, turn the divide sign into a multiply and turn the second fraction upside down. This is known as multiplying by the reciprocal. The sum then becomes multiplying two fractions, which is done using the method above.

Multiplying and dividing rational expressions works using the same methods.

Multiplying rational expressions

To multiply two rational expressions, multiply the numerators and denominators together.

Example

Simplify \frac{5m}{6m^2} \times \frac{3}{4m}.

5m \times 3 = 15m

6m^2 \times 4m = 24m^3

\frac{5m}{6m^2} \times \frac{3}{4m} = \frac{15m}{24m^3}

This fraction can be simplified by dividing common factors. 15m and 24m^3 have a highest common factor of 3m.

\frac{15m \div 3m}{24m^3 \div 3m} = \frac{5}{8m^2}

The fraction now has no further common factors, so this is the final answer.

This fraction could also be simplified by removing any common factors before multiplying together. To simplify the fraction in this way, first check all numerators and denominators and cancel out any common factors.

There is a common factor of 3 and m between the numerators and denominators.

\frac{5 \cancel{m}}{2 \cancel{6} m^2} \times \frac{1 \cancel{3}}{4 \cancel{m}} = \frac{5}{2m^2} \times \frac{1}{4} = \frac{5 \times 1}{2m^2  \times 4} = \frac{5}{8m^2}

Note that the answer is the same regardless of whether common factors are divided first or last.

Dividing rational expressions

To divide two algebraic expressions, keep the first fraction the same, turn the division sign into a multiplication sign and turn the second fraction upside down.

Example

Simplify \frac{7p}{2} \div \frac{1}{3p}.

\frac{7p}{2} \div \frac{1}{3p} = \frac{7p}{2} \times \frac{3p}{1} = \frac{7p \times 3p}{2 \times 1} = \frac{21p^2}{2}

There are no common factors so this is the final answer.

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