Complex Rational Expressions

Learning Outcomes

• Simplify complex rational expressions

Fractions and rational expressions can be interpreted as quotients. When both the dividend (numerator) and divisor (denominator) include fractions or rational expressions, you have something more complex than usual. Do not fear—you have all the tools you need to simplify these quotients!

A complex fraction is the quotient of two fractions. These complex fractions are never considered to be in simplest form, but they can always be simplified using division of fractions. Remember, to divide fractions, you multiply by the reciprocal.

Before you multiply the numbers, it is often helpful to factor the fractions. You can then cancel factors.

Example

Simplify.

$\displaystyle\dfrac{\,\frac{12}{35}\,}{\,\frac{6}{7}\,}$

If two fractions appear in the numerator or denominator (or both), first combine them. Then simplify the quotient as shown above.

Example

Simplify.

$\displaystyle\Large \frac{\,\frac{3}{4}+\frac{1}{2}\,}{\,\frac{4}{5}-\frac{1}{10}\,}$

In the following video, we will show a couple more examples of how to simplify complex fractions.

Complex Rational Expressions

A complex rational expression is a quotient with rational expressions in the dividend, divisor, or in both. Simplify these in the exact same way as you would a complex fraction.

Example

Simplify.

$\displaystyle\Large \frac{\,\,\frac{x+5}{{{x}^{2}}-16}\,}{\,\,\frac{{{x}^{2}}-\,\,25}{x-4}\,}$

In the next video example, we will show that simplifying a complex fraction may require factoring first.

The same ideas can be used when simplifying complex rational expressions that include more than one rational expression in the numerator or denominator. However, there is a shortcut that can be used. Compare these two examples of simplifying a complex fraction.

Example

Simplify.

$\displaystyle\dfrac{\,\,\normalsize1-\dfrac{9}{{{x}^{2}}}\,\,}{\,\,\normalsize1+\dfrac{5}{x}\normalsize+\dfrac{6}{{{x}^{2}}}\,\,}$

Example

Simplify.

$\frac{1-\frac{9}{{{x}^{2}}}}{1+\frac{5}{x}+\frac{6}{{{x}^{2}}}}$

You may find the second method easier to use, but do try both ways to see what you prefer.

In our last example, we show a similar example as the one above.

Summary

Complex rational expressions are quotients with rational expressions in the divisor, dividend, or both. When written in fraction form, they appear to be fractions within a fraction. These can be simplified by first treating the quotient as a division problem. Then you can rewrite the division as multiplication and take the reciprocal of the divisor. Or you can simplify the complex rational expression by multiplying both the numerator and denominator by a denominator common to all rational expressions within the complex expression. This can help simplify the complex expression even faster.