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.