Simplifying rational expressions

Simplifying rational expressions or algebraic fractions works in the same way as simplifying normal fractions. A common factor must be found and divided throughout. For example, to simplify the fraction \frac{12}{16}, look for a common factor between 12 and 16. This is 4 as 4 \times 3 = 12 and 4 \times 4 = 16.

Divide 4 throughout the fraction, which gives \frac{12 \div 4}{16 \div 4} = \frac{3}{4}.

Example 1

Simplify \frac{6m^2}{3m}.

To simplify this, look for the highest common factor of 6m^2 and 3m. This is 3m. Take this common factor out of each part of the fraction.

This gives \frac{6m^2 \div 3m}{3m \div 3m} = \frac{2m}{1} = 2m.

This fraction cannot be simplified any further so this is the final answer.

Question

Simplify \frac{4(p + 7)}{(p + 7)^2}.

The highest common factor of 4(p + 7) and (p + 7)^2 is (p + 7). Divide this common factor through the numerator and denominator.

This gives \frac{4(p + 7) \div (p + 7)}{(p + 7)^2 \div (p + 7)} = \frac{4}{p + 7}.

This fraction cannot be simplified any further so this is the final answer.

Question

Simplify \frac{(m - 7)(m + 3)}{6(m + 3)}.

The highest common factor of (m - 7)(m + 3) and 6(m + 3) is (m + 3). Dividing this throughout the fraction gives \frac{(m - 7)(m + 3) \div (m + 3)}{6(m + 3) \div (m + 3)} = \frac{m - 7}{6}.

There are no more common factors in this fraction, so this is the final answer.