7.3 Simplify Complex Rational Expressions

Topics covered in this section are:

Simplify a complex rational expression by writing it as division
Simplify a complex rational expression by using the LCD

7.3.1 Simplify a Complex Rational Expression by Writing it as Division

Complex fractions are fractions in which the numerator or denominator contains a fraction. We previously simplified complex fractions like these:

$\frac{\frac{3}{4}}{\frac{5}{8}}$ $\frac{\frac{x}{2}}{\frac{xy}{6}}$

In this section, we will simplify complex rational expressions, which are rational expressions with rational expressions in the numerator or denominator.

COMPLEX RATIONAL EXPRESSION

A complex rational expression is a rational expression in which the numerator and/or the denominator contains a rational expression.

Here are a few complex rational expressions:

$\frac{ \frac{4}{y-3}} {\frac{8}{y^{2}-9}}$

$\frac{ \frac{1}{x}+\frac{1}{y}} {\frac{x}{y}-\frac{y}{x}}$

$\frac{ \frac{2}{x+6}} {\frac{4}{x-6}-\frac{4}{x^{2}-36}}$

Remember, we always exclude values that would make any denominator zero.

We will use two methods to simplify complex rational expressions.

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

$\frac{ \frac{6x^{2}-7x+2}{4x-8}} { \frac{2x^{2}-8x+3}{x^{2}-5x+6}}$

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

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

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 complex rational expressions. We make sure the complex rational expression is of the form where one fraction is over one fraction. We then write it as if we were dividing two fractions.

Example 1

Simplify the complex expression by writing it as division: $\frac{ \frac{6}{x-4}} {\frac{3}{x^{2}-16}}$.

Solution

	$\frac{ \frac{6}{x-4}} {\frac{3}{x^{2}-16}}$
Rewrite the complex fraction as division.	$ \frac{6}{x-4} \div \frac{3}{x^{2}-16}$
Rewrite as the product of the first times the reciprocal of the second.	$ \frac{6}{x-4} \cdot \frac{x^{2}-16}{3}$
Factor.	$ \frac{3 \cdot 2}{x-4} \cdot \frac{(x-4)(x+4)}{3}$
Multiply.	$\frac{3 \cdot 2(x-4)(x+4)}{3(x-4)}$
Remove common factors.	$\frac{\cancel{3} \cdot 2\cancel{(x-4)}(x+4)}{\cancel{3}\cancel{(x-4)}}$
Simplify.	$2(x+4)$

Are there any values of $x$ that should not be allowed? The original complex rational expression had denominators of $x-4$ and $x^{2}-16$. This expression would be undefined if $x=4$ or $x=-4$.

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 2

Simplify the complex rational expression by writing it as division: $\frac{ \frac{1}{3} + \frac{1}{6}}{ \frac{1}{2} – \frac{1}{3}}$.

Solution

	$\frac{ \frac{1}{3} + \frac{1}{6}}{ \frac{1}{2} – \frac{1}{3}}$
Simplify the numerator and denominator. Find the LCD and add the fractions in the numerator. Find the LCD and subtract the fractions in the denominator.	$\frac{ \frac{1 \cdot \textcolor{red}{2}}{3 \cdot \textcolor{red}{2}}+ \frac{1}{6}}{ \frac{1 \cdot \textcolor{red}{3}}{2 \cdot \textcolor{red}{3}} – \frac{1 \cdot \textcolor{red}{2}}{3 \cdot \textcolor{red}{2}}}$
Simplify the numerator and denominator.	$\frac{ \frac{2}{6} + \frac{1}{6}}{ \frac{3}{6} – \frac{2}{6}}$
Rewrite the complex rational expression as a division problem.	$\frac{3}{6} \div \frac{1}{6}$
Multiply the first by the reciprocal of the second.	$\frac{3}{6} \cdot \frac{6}{1}$
Simplify.	$3$

We follow the same procedure when the complex rational expression contains variables.

Example 3

Simplify the complex rational expression by writing it as division: $\frac{ \frac{1}{x} + \frac{1}{y}}{ \frac{x}{y} – \frac{y}{x}}$.

Solution

Step 1 is to simplify the sum in the numerator and the difference in the denominator of complex rational expression, the quantity 1 divided by x plus 1 divided by y all divided by the quantity x divided by y minus y divided by x. The common denominator of the fractions in the complex rational expression is x y. Multiply the numerator and denominator of 1 divided by x by y over y. Multiply the numerator and denominator of 1 divided by y by x over x. Multiply the numerator and denominator of x divided by y by x over x. Multiply the numerator and denominator of y over x by y over y. The result is the quantity y divided by x y plus x divided by x y all divided by the quantity x squared divided by x y minus y squared divided by x y. Add the fractions in the numerator and subtract the fractions in the denominator. The result is the sum of y and x divided by x y all divided by the difference between x squared and y squared divided by x y. We now have just one rational expression in the numerator and one in the denominator.

Step 2 is to rewrite the complex rational expression as a division problem. Write the numerator divided by the denominator. The result is the quantity of the sum y and x divided by x y all divided by the quantity of the difference between x squared and y squared divided by x y.

Step 3 is to divided the expressions. Multiply the first expression by the reciprocal of the second expression. The result is the quantity of the sum y and x divided by x y times the quantity x y divided by the difference between x squared and y squared. Factor any expressions if possible. The result is the product of x y and the sum of y and x all divided by the product of x y, the difference between x and y, and the sum of x and y. Remove the common factors, x y and the sum of x and y. Simplify. The result is 1 divided by the quantity x minus y.

We summarize the steps here.

HOW TO: 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 4

Simplify the complex rational expression by writing it as division: $\frac{ n – \frac{4n}{n+5}}{ \frac{1}{n+5} + \frac{1}{n-5}}$.

Solution

	$\frac{ n – \frac{4n}{n+5}}{ \frac{1}{n+5} + \frac{1}{n-5}}$
Simplify the numerator and denominator. Find common denominators for the numerator and denominator.	$\frac{ \frac{n \textcolor{red}{(n+5)}}{1 \textcolor{red}{(n+5)}}- \frac{4n}{n+5}}{ \frac{1 \textcolor{red}{(n-5)}}{(n+5)\textcolor{red}{(n-5)}} + \frac{1\textcolor{red}{(n+5)}}{(n-5) \textcolor{red}{(n+5)}}}$
Simplify the numerators.	$\frac{ \frac{n^{2}+5n}{n+5}- \frac{4n}{n+5}}{ \frac{n-5}{(n+5)(n-5)} + \frac{n+5}{(n-5)(n+5)}}$
Subtract the rational expressions in the numerator and add in the denominator.	$\frac{ \frac{n^{2}+5n-4n}{n+5}}{ \frac{n-5+n+5}{(n+5)(n-5)}}$
Simplify. (We now have one rational expression over one rational expression).	$\frac{ \frac{n^{2}+n}{n+5}}{ \frac{2n}{(n+5)(n-5)}}$
Rewrite as fraction division.	$ \frac{n^{2}+n}{n+5} \div \frac{2n}{(n+5)(n-5)}$
Multiply the first times the reciprocal of the second.	$ \frac{n^{2}+n}{n+5} \cdot \frac{(n+5)(n-5)}{2n}$
Factor any expressions if possible.	$ \frac{n(n+1)(n+5)(n-5)}{(n+5)2n}$
Remove common factors.	$ \frac{\cancel{n}(n+1)\cancel{(n+5)}(n-5)} {\cancel{(n+5)}2\cancel{n}}$
Simplify.	$ \frac{(n+1)(n-5)}{2}$

7.3.2 Simplify a Complex Rational Expression by Using the LCD

We “cleared” the fractions by multiplying by the LCD when we solved equations with fractions. We can use that strategy here to simplify complex rational expressions. We will multiply the numerator and denominator by the LCD of all the rational expressions.

Let’s look at the complex rational expression we simplified one way in Example 2. We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by $\frac{\text{LCD}}{\text{LCD}}$ we are multiplying by $1$, so the value stays the same.

Example 5

Simplify the complex rational expression by using the LCD: $\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}$.

Solution

	$\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}$
The LCD of all the fractions in the whole expression is $6$.
Clear the fractions by multiplying the numerator and the denominator by that LCD.	$\frac{\textcolor{red}{6} \cdot (\frac{1}{3}+\frac{1}{6})}{\textcolor{red}{6} \cdot (\frac{1}{2}-\frac{1}{3})}$
Distribute.	$\frac{\textcolor{red}{6} \cdot \frac{1}{3}+\textcolor{red}{6} \cdot \frac{1}{6}}{\textcolor{red}{6} \cdot \frac{1}{2}-\textcolor{red}{6} \cdot \frac{1}{3}}$
Simplify.	$\frac{2+1}{3-2}$
	$\frac{3}{1}$
	$3$

We will use the same example as in Example 3. Decide which method works better for you.

Example 6

Simplify the complex rational expression by using the LCD: $\frac{ \frac{1}{x} + \frac{1}{y}}{ \frac{x}{y} – \frac{y}{x}}$.

Solution

Step 1 is to find the least common denominator of the complex rational expression, the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by the difference between the quantity x divided by y and the quantity y divided by x.

Step 2 is to multiply the numerator and the denominator by the least common denominator, x y. The result is x y times the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by x y times the difference between the quantity x divided by y and the quantity y divided by x.

Step 3 is to simplify the expression. Distribute x y in the numerator and the denominator. The result is x y times 1 divided by x plus x y times 1 divided by y all divided by x y times x divided by y plus x y times y divided by x. It simplifies to the sum of y and x divided by the quantity x squared minus y squared. Write the denominator as the difference of squares, the quantity x minus y times the quantity x plus y. The result is the quantity y plus x all divided by the quantity x minus y times the quantity x plus y. Remove the common factor, y plus x, from the numerator and denominator. The result is 1 divided by the quantity x minus y.

HOW TO: Simplify a complex rational expression by using the LCD.

Find the LCD of all fractions in the complex rational expression.
Multiply the numerator and denominator by the LCD.
Simplify the expression.

Be sure to start by factoring all the denominators so you can find the LCD.

Example 7

Simplify the complex rational expression by using the LCD: $\frac{ \frac{2}{x+6}}{ \frac{4}{x-6} – \frac{4}{x^{2}-36}}$.

Solution

	$\frac{ \frac{2}{x+6}}{ \frac{4}{x-6} – \frac{4}{x^{2}-36}}$
Find the LCD of all the fractions in the complex rational expression. The LCD is $x^{2}-36= (x+6)(x-6)$.
Multiply the numerator and denominator by the LCD.	$\frac{\textcolor{red}{(x+6)(x-6)} \frac{2}{x+6}}{\textcolor{red}{(x+6)(x-6)} \left(\frac{4}{x-6}-\frac{4}{(x+6)(x-6)}\right)}$
Simplify the expression.
Distribute the denominator.	$\frac{(x+6)(x-6)\frac{2}{x+6}}{\textcolor{red}{(x+6)(x-6)} \left(\frac{4}{x-6}\right) \textcolor{red}{-(x+6)(x-6)} \left(\frac{4}{(x+6)(x-6)}\right)}$
Remove common factors.	$\frac{\cancel{(x+6)}(x-6)\frac{2}{\cancel{x+6}}}{\textcolor{red}{(x+6)\cancel{(x-6)}} \left(\frac{4}{\cancel{x-6}}\right) \textcolor{red}{-\cancel{(x+6)(x-6)}} \left(\frac{4}{\cancel{(x+6)(x-6)}}\right)}$
Simplify.	$\frac{2(x-6)}{4(x+6)-4}$
To simplify the denominator, distribute and combine like terms.	$\frac{2(x-6)}{4x+20}$
Factor the denominator.	$\frac{2(x-6)}{4(x+5)}$
Remove common factors.	$\frac{\cancel{2}(x-6)}{\cancel{2}\cdot 2(x+5)}$
Simplify.	$\frac{x-6}{2(x+5)}$
	Notice that there are no more factors common to the numerator and denominator.

Be sure to factor the denominators first. Proceed carefully as the math can get messy!

Example 8

Simplify the complex rational expression by using the LCD: $\frac{ \frac{4}{m^{2}-7m+12}}{ \frac{3}{m-3} – \frac{2}{m-4}}$.

Solution

	$\frac{ \frac{4}{m^{2}-7m+12}}{ \frac{3}{m-3} – \frac{2}{m-4}}$
Find the LCD of all the fractions in the complex rational expression. The LCD is $(m-3)(m-4)$.
Multiply the numerator and denominator by the LCD.	$\frac{\textcolor{red}{(m-3)(m-4)} \frac{4}{(m-3)(m-4)}}{\textcolor{red}{(m-3)(m-4)} \left(\frac{3}{m-3}-\frac{2}{(m-4)}\right)}$
Distribute the denominator and remove common factors.	$\frac{\cancel{\textcolor{red}{(m-3)(m-4)}} \frac{4}{\cancel{(m-3)(m-4)}}}{\textcolor{red}{\cancel{(m-3)}(m-4)} \left(\frac{3}{\cancel{m-3}}\right)-\textcolor{red}{(m-3)\cancel{(m-4)}}\left(\frac{2}{\cancel{(m-4)}}\right)}$
Simplify.	$\frac{4}{3(m-4)-2(m-3)}$
Distribute.	$\frac{4}{3m-12-2m+6}$
Combine like terms.	$\frac{4}{m-6}$

Example 9

Simplify the complex rational expression by using the LCD: $\frac{ \frac{y}{y+1}}{ 1 + \frac{1}{y-1}}$.

Solution

	$\frac{ \frac{y}{y+1}}{ 1 + \frac{1}{y-1}}$
Find the LCD of all the fractions in the complex rational expression. The LCD is $(y+1)(y-1)$.
Multiply the numerator and denominator by the LCD.	$\frac{\textcolor{red}{(y+1)(y-1)} \frac{y}{y+1}}{\textcolor{red}{(y+1)(y-1)} \left(1+\frac{1}{y-1}\right)}$
Distribute the denominator and simplify.	$\frac{\textcolor{red}{\cancel{(y+1)}(y-1)} \frac{y}{\cancel{y+1}}}{\textcolor{red}{(y+1)(y-1)}(1)+ \textcolor{red}{(y+1)\cancel{(y-1)}}\left(1+\frac{1}{\cancel{y-1}}\right)}$
Simplify.	$\frac{(y-1)y}{(y+1)(y-1)+(y+1)}$
Simplify the denominator and leave the numerator factored.	$\frac{y(y-1)}{y^{2}-1+y+1}$
	$\frac{y(y-1)}{y^{2}+y}$
Factor the denominator and remove factors common with the numerator.	$\frac{\cancel{y}(y-1)}{\cancel{y}(y+1)}$
Simplify.	$\frac{y-1}{y+1}$