Mathematics » Rational Expressions and Equations » Simplify Complex Rational Expressions

Simplifying a Complex Rational Expression By Writing It As Division

Simplifying a Complex Rational Expression By Writing It As Division

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

The above image has three columns. The image shows steps on how to divide complex rational expressions in three steps. Step one is to simplify the numerator and denominator. We will simplify the sum in the numerator and difference in the denominator for the example 1 divided by x plus 1 divided by y divided by x divided by y minus y divided by x. Find a common denominator and add the fractions in the numerator and find a common denominator and subtract the fractions in the numerator to get 1 times y divided by x times y plus 1 times x divided by y times x divided by x times x divided by y times x minus y times y divided by x times y. Then, we get y divided by x y plus x plus x y divided by x squared divided by x y minus y squared divided by x y. We now have just one rational expression in the numerator and one in the denominator, y plus x divided by x y divided by x squared minus y squared divided by x y.Step two is to rewrite the complex rational expression as a division problem. We write the numerator divided by the denominator.Step three is to divide the expressions. Multiply the first by the reciprocal of the second to get y plus x divided by x y times x y divided by x squared minus y squared. Factor any expressions if possible. We now have x y times y plus x divided by x y times x minus y times x plus y. Remove common factors. Cross out x, y and y plus x from the numerator. Cross out x, y and x plus y from the denominator. Simplify to get 1 divided by x minus y.

Simplify a complex rational expression by writing it as division.

  1. Simplify the numerator and denominator.
  2. Rewrite the complex rational expression as a division problem.
  3. 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..

[Attributions and Licenses]


This is a lesson from the tutorial, Rational Expressions and Equations and you are encouraged to log in or register, so that you can track your progress.

Log In

Share Thoughts