# Simplifying rational expressions

Simplifying expressions or algebraic fractions works in the same way as simplifying normal fractions. A common 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 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 and .

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.