## Simplifying a Complex Rational Expression By Writing It As Division

Contents

We have already seen this complex rational expression earlier in this tutorial.

\(\cfrac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-8x+3}{{x}^{2}-5x+6}}\)

We noted that fraction bars tell us to divide, so rewrote it as the division problem

\(\left(\frac{6{x}^{2}-7x+2}{4x-8}\right)÷\left(\frac{2{x}^{2}-8x+3}{{x}^{2}-5x+6}\right)\)

Then we multiplied the first rational expression by the reciprocal of the second, just like we do when we divide two fractions.

This is one method to simplify rational expressions. We write it as if we were dividing two fractions.

## Example

Simplify: \(\cfrac{\frac{4}{y-3}}{\frac{8}{{y}^{2}-9}}.\)

### Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\cfrac{\frac{4}{y-3}}{\frac{8}{{y}^{2}-9}}\hfill \\ \\ \text{Rewrite the complex fraction as division.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4}{y-3}÷\frac{8}{{y}^{2}-9}\hfill \\ \\ \begin{array}{c}\text{Rewrite as the product of first times the}\hfill \\ \text{reciprocal of the second.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4}{y-3}·\frac{{y}^{2}-9}{8}\hfill \\ \\ \text{Multiply.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4\left({y}^{2}-9\right)}{8\left(y-3\right)}\hfill \\ \\ \text{Factor to look for common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4\left(y-3\right)\left(y+3\right)}{4·2\left(y-3\right)}\hfill \\ \\ \text{Remove common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\require{cancel}\cancel{4}\require{cancel}\cancel{\left(y-3\right)}\left(y+3\right)}{\require{cancel}\cancel{4}·2\require{cancel}\cancel{\left(y-3\right)}}\hfill \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{y+3}{2}\hfill \end{array}\)

Are there any value(s) of \(y\) that should not be allowed? The simplified rational expression has just a constant in the denominator. But the original **complex rational expression** had denominators of \(y-3\) and \({y}^{2}-9\). This expression would be undefined if \(y=3\) or \(y=-3\).

Fraction bars act as grouping symbols. So to follow the Order of Operations, we simplify the numerator and denominator as much as possible before we can do the division.

## Example

Simplify: \(\cfrac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}.\)

### Solution

Simplify the numerator and denominator. | |

Find the LCD and add the fractions in the numerator. Find the LCD and add the fractions in the denominator. | |

Simplify the numerator and denominator. | |

Simplify the numerator and denominator, again. | |

Rewrite the complex rational expression as a division problem. | |

Multiply the first times by the reciprocal of the second. | |

Simplify. |

### Example: How to Simplify a Complex Rational Expression by Writing it as Division

Simplify: \(\cfrac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}.\)

### Solution

### Simplify a complex rational expression by writing it as division.

- Simplify the numerator and denominator.
- Rewrite the complex rational expression as a division problem.
- Divide the expressions.

## Example

Simplify: \(\frac{n-\frac{4n}{n+5}}{\frac{1}{n+5}+\frac{1}{n-5}}.\)

### Solution

Simplify the numerator and denominator. | |

Find the LCD and add the fractions in the numerator. Find the LCD and add the fractions in the denominator. | |

Simplify the numerators. | |

Subtract the rational expressions in the numerator andadd in the denominator. Simplify. | |

Rewrite as fraction division. | |

Multiply the first times the reciprocal of the second. | |

Factor any expressions if possible. | |

Remove common factors. | |

Simplify. |