Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

15.4: Simplify Complex Rational Expressions

( \newcommand{\kernel}{\mathrm{null}\,}\)

Learning Objectives

By the end of this section, you will be able to:

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

Before you get started, take this readiness quiz.

  1. Simplify: 35910.
    If you missed this problem, review [link].
  2. Simplify: 11342+4·5.
    If you missed this problem, review [link].
  3. Solve: 12x+14=18.
    If you missed this problem, review [link].

Complex Fraction

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

We previously simplified complex fractions like these:

3458x2xy6

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:

4y38y291x+1yxyyx2x+64x64x236

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.

6x27x+24x82x28x+3x25x+6

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

(6x27x+24x8)÷(2x28x+3x25x+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 15.4.1

Simplify the complex rational expression by writing it as division: 6x43x216

Solution

Rewrite the complex fraction as division. 6x4÷3x216

Rewrite as the product of first times the reciprocal of the second.Rewrite as the product of first times the reciprocal of the second.Rewrite as the product of first times the reciprocal of the second.

6x4x2163

Factor.

32x4(x4)(x+4)3

Multiply.

32(x4)(x+4)3(x4)

Remove common factors.

32(x4)(x+4)3(x4)

Simplify.

2(x+4)

Are there any value(s) of x that should not be allowed? The original complex rational expression had denominators of x4 and x216. This expression would be undefined if x=4 or x=-4.

Try It \PageIndex{1}

Simplify the complex rational expression by writing it as division: \dfrac{\dfrac{2}{x^{2}-1}}{\dfrac{3}{x+1}} \nonumber

Answer

\dfrac{2}{3(x-1)}

Try It \PageIndex{2}

Simplify the complex rational expression by writing it as division: \dfrac{\dfrac{1}{x^{2}-7 x+12}}{\dfrac{2}{x-4}} \nonumber

Answer

\dfrac{1}{2(x-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 \PageIndex{2}

Simplify the complex rational expression by writing it as division: \dfrac{\dfrac{1}{3}+\dfrac{1}{6}}{\dfrac{1}{2}-\dfrac{1}{3}} \nonumber

Solution

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.

\dfrac{\dfrac{1 \cdot {\color{red}2}}{3 \cdot {\color{red}2}}+\dfrac{1}{6}}{\dfrac{1 \cdot {\color{red}3}}{2 \cdot {\color{red}3}}-\dfrac{1 \cdot {\color{red}2}}{3 \cdot {\color{red}2}}} \nonumber

Simplify the numerator and denominator.

\dfrac{\dfrac{2}{6}+\dfrac{1}{6}}{\dfrac{3}{6}-\dfrac{2}{6}} \nonumber

Rewrite the complex rational expression as a division problem.

\dfrac{3}{6} \div \dfrac{1}{6} \nonumber

Multiply the first by the reciprocal of the second.

\dfrac{3}{6} \cdot \dfrac{6}{1} \nonumber

Simplify.

3 \nonumber

Try It \PageIndex{3}

Simplify the complex rational expression by writing it as division: \dfrac{\dfrac{1}{2}+\dfrac{2}{3}}{\dfrac{5}{6}+\dfrac{1}{12}} \nonumber

Answer

\dfrac{14}{11}

Try It \PageIndex{4}

Simplify the complex rational expression by writing it as division: \dfrac{\dfrac{3}{4}-\dfrac{1}{3}}{\dfrac{1}{8}+\dfrac{5}{6}} \nonumber

Answer

\dfrac{10}{23}

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

Example \PageIndex{3}: How to Simplify a Complex Rational Expression using Division

Simplify the complex rational expression by writing it as division: \dfrac{\dfrac{1}{x}+\dfrac{1}{y}}{\dfrac{x}{y}-\dfrac{y}{x}} \nonumber

Solution

Step 1. Simplify the numerator.

We will simplify the sum in and denominator. the numerator and difference in the denominator.

\dfrac{\dfrac{1}{x}+\dfrac{1}{y}}{\dfrac{x}{y}-\dfrac{y}{x}} \nonumber

Find a common denominator and add the fractions in the numerator.

\dfrac{\dfrac{1 \cdot {\color{red}y}}{x \cdot {\color{red}y}}+\dfrac{1 \cdot {\color{red}x}}{y \cdot {\color{red}x}}}{\dfrac{x \cdot {\color{red}x}}{y \cdot {\color{red}x}}-\dfrac{y \cdot {\color{red}y}}{x \cdot {\color{red}y}}} \nonumber

\dfrac{\dfrac{y}{x y}+\dfrac{x}{x y}}{\dfrac{x^{2}}{x y}-\dfrac{y^{2}}{x y}} \nonumber

Find a common denominator and subtract the fractions in the denominator.

\dfrac{\dfrac{y+x}{x y}}{\dfrac{x^{2}-y^{2}}{x y}} \nonumber

We now have just one rational expression in the numerator and one in the denominator.

Step 2. Rewrite the complex rational expression as a division problem.

We write the numerator divided by the denominator.

\left(\dfrac{y+x}{x y}\right) \div\left(\dfrac{x^{2}-y^{2}}{x y}\right) \nonumber

Step 3. Divide the expressions.

Multiply the first by the reciprocal of the second.

\left(\dfrac{y+x}{x y}\right) \cdot\left(\dfrac{x y}{x^{2}-y^{2}}\right) \nonumber

Factor any expressions if possible.

\dfrac{x y(y+x)}{x y(x-y)(x+y)} \nonumber

Remove common factors.

\dfrac{\cancel {x y}\cancel {(y+x)}}{\cancel {x y}(x-y)\cancel {(x+y)}} \nonumber

Simplify.

\dfrac{1}{x-y} \nonumber

Try It \PageIndex{5}

Simplify the complex rational expression by writing it as division: \dfrac{\dfrac{1}{x}+\dfrac{1}{y}}{\dfrac{1}{x}-\dfrac{1}{y}} \nonumber

Answer

\dfrac{y+x}{y-x}

Try It \PageIndex{6}

Simplify the complex rational expression by writing it as division: \dfrac{\dfrac{1}{a}+\dfrac{1}{b}}{\dfrac{1}{a^{2}}-\dfrac{1}{b^{2}}} \nonumber

Answer

\dfrac{a b}{b-a}

We summarize the steps here.

How to simplify a complex rational expression by writing it as division.
  1. Rewrite the complex rational expression as a division problem.
  2. Divide the expressions.
Example \PageIndex{4}

Simplify the complex rational expression by writing it as division: \dfrac{n-\dfrac{4 n}{n+5}}{\dfrac{1}{n+5}+\dfrac{1}{n-5}} \nonumber

Solution

Simplify the numerator and denominator. Find common denominators for the numerator and denominator.

\dfrac{\dfrac{n{\color{red}(n+5)}}{1{\color{red}(n+5)}}-\dfrac{4 n}{n+5}}{\dfrac{1{\color{red}(n-5)}}{(n+5){\color{red}(n-5)}}+\dfrac{1{\color{red}(n+5)}}{(n-5){\color{red}(n+5)}}} \nonumber

Simplify the numerators.

\dfrac{\dfrac{n^{2}+5 n}{n+5}-\dfrac{4 n}{n+5}} {\dfrac{n-5}{(n+5)(n-5)}+\dfrac{n+5}{(n-5)(n+5)}} \nonumber

Subtract the rational expressions in the numerator and add in the denominator.

\dfrac{\dfrac{n^{2}+5 n-4 n}{n+5}}{\dfrac{n-5+n+5}{(n+5)(n-5)}} \nonumber

Simplify. (We now have one rational expression over one rational expression.)

\dfrac{\dfrac{n^{2}+n}{n+5}}{\dfrac {2n}{(n+5)(n-5)}} \nonumber

Rewrite as fraction division.

\dfrac{n^{2}+n}{n+5} \div \dfrac{2 n}{(n+5)(n-5)} \nonumber

Multiply the first times the reciprocal of the second.

\dfrac{n^{2}+n}{n+5} \cdot \dfrac{(n+5)(n-5)}{2 n} \nonumber

Factor any expressions if possible.

\dfrac{n(n+1)(n+5)(n-5)}{(n+5) 2 n} \nonumber

Remove common factors.

\dfrac{\cancel{n}(n+1)\cancel {(n+5)}(n-5)}{\cancel {(n+5)} 2 \cancel {n}} \nonumber

Simplify.

\dfrac{(n+1)(n-5)}{2} \nonumber

Try It \PageIndex{7}

Simplify the complex rational expression by writing it as division: \dfrac{b-\dfrac{3 b}{b+5}}{\dfrac{2}{b+5}+\dfrac{1}{b-5}} \nonumber

Answer

\dfrac{b(b+2)(b-5)}{3 b-5}

Try It \PageIndex{8}

Simplify the complex rational expression by writing it as division: \dfrac{1-\dfrac{3}{c+4}}{\dfrac{1}{c+4}+\dfrac{c}{3}} \nonumber

Answer

\dfrac{3}{c+3}

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 7.4.2. We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by \dfrac{LCD}{LCD} we are multiplying by 1, so the value stays the same.

Example \PageIndex{5}

Simplify the complex rational expression by using the LCD: \dfrac{\dfrac{1}{3}+\dfrac{1}{6}}{\dfrac{1}{2}-\dfrac{1}{3}} \nonumber

Solution

The LCD of all the fractions in the whole expression is 6.

Clear the fractions by multiplying the numerator and denominator by that LCD.

\dfrac{{\color{red}6} \cdot\left(\dfrac{1}{3}+\dfrac{1}{6}\right)}{{\color{red}6} \cdot\left(\dfrac{1}{2}-\dfrac{1}{3}\right)} \nonumber

Distribute.

\dfrac{6 \cdot \dfrac{1}{3}+6 \cdot \dfrac{1}{6}}{6 \cdot \dfrac{1}{2}-6 \cdot \dfrac{1}{3}} \nonumber

Simplify.

\dfrac{2+1}{3-2} \nonumber

\dfrac{3}{1}\nonumber

3\nonumber

Try It \PageIndex{9}

Simplify the complex rational expression by using the LCD: \dfrac{\dfrac{1}{2}+\dfrac{1}{5}}{\dfrac{1}{10}+\dfrac{1}{5}} \nonumber

Answer

\dfrac{7}{3}

Try It \PageIndex{10}

Simplify the complex rational expression by using the LCD: \dfrac{\dfrac{1}{4}+\dfrac{3}{8}}{\dfrac{1}{2}-\dfrac{5}{16}} \nonumber

Answer

\dfrac{10}{3}

We will use the same example as in Example 7.4.3. Decide which method works better for you.

Example \PageIndex{6}: How to Simplify a Complex Rational Expressing using the LCD

Simplify the complex rational expression by using the LCD: \dfrac{\dfrac{1}{x}+\dfrac{1}{y}}{\dfrac{x}{y}-\dfrac{y}{x}} \nonumber

Solution

Step 1. Find the LCD of all fractions in the is complex rational expression.

The LCD of all the fractions xy.

\dfrac{\dfrac{1}{x}+\dfrac{1}{y}}{\dfrac{x}{y}-\dfrac{y}{x}} \nonumber

Step 2. Multiply the numerator and denominator by the LCD.

Multiply both the numerator and denominator by xy.

\dfrac{{\color{red}x y} \cdot\left(\dfrac{1}{x}+\dfrac{1}{y}\right)}{{\color{red}x y} \cdot\left(\dfrac{x}{y}-\dfrac{y}{x}\right)} \nonumber

Step 3. Simplify the expression.

Distribute.

\dfrac{xy \cdot \dfrac{1}{x}+xy \cdot \dfrac{1}{y}}{xy \cdot \dfrac{x}{y}-xy \cdot \dfrac{y}{x}} \nonumber

\dfrac{y+x}{x^{2}-y^{2}} \nonumber

Simplify.

\dfrac{\cancel{(y+x)}}{(x-y)\cancel{(x+y)}} \nonumber

Remove common factors.

\dfrac{1}{x-y} \nonumber

Try It \PageIndex{11}

Simplify the complex rational expression by using the LCD: \dfrac{\dfrac{1}{a}+\dfrac{1}{b}}{\dfrac{a}{b}+\dfrac{b}{a}} \nonumber

Answer

\dfrac{b+a}{a^{2}+b^{2}}

Try It \PageIndex{12}

Simplify the complex rational expression by using the LCD: \dfrac{\dfrac{1}{x^{2}}-\dfrac{1}{y^{2}}}{\dfrac{1}{x}+\dfrac{1}{y}} \nonumber

Answer

\dfrac{y-x}{x y}

How to simplify a complex rational expression by using the LCD.
  1. Multiply the numerator and denominator by the LCD.
  2. Simplify the expression.

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

Example \PageIndex{7}

Simplify the complex rational expression by using the LCD: \dfrac{\dfrac{2}{x+6}}{\dfrac{4}{x-6}-\dfrac{4}{x^{2}-36}} \nonumber

Solution

Find the LCD of all fractions in the complex rational expression. The LCD is:

x^{2}-36=(x+6)(x-6) \nonumber

Multiply the numerator and denominator by the LCD.

\dfrac{(x+6)(x-6) \dfrac{2}{x+6}}{(x+6)(x-6)\left(\dfrac{4}{x-6}-\dfrac{4}{(x+6)(x-6)}\right)} \nonumber

Simplify the expression.

Distribute in the denominator.

\dfrac{(x+6)(x-6) \dfrac{2}{x+6}}{{\color{red}(x+6)(x-6)}\left(\dfrac{4}{x-6}\right)-{\color{red}(x+6)(x-6)}\left(\dfrac{4}{(x+6)(x-6)}\right)} \nonumber

Simplify.

\dfrac{\cancel{(x+6)}(x-6) \dfrac{2}{\cancel{x+6}}}{{\color{red}(x+6)\cancel{(x-6)}}\left(\dfrac{4}{x-6}\right)-{\color{red}\cancel{(x+6)(x-6)}}\left(\dfrac{4}{\cancel{(x+6)(x-6)}}\right)} \nonumber

Simplify.

\dfrac{2(x-6)}{4(x+6)-4} \nonumber

To simplify the denominator, distribute and combine like terms.

\dfrac{2(x-6)}{4 x+20} \nonumber

Factor the denominator.

\dfrac{2(x-6)}{4(x+5)} \nonumber

Remove common factors.

\dfrac{\cancel{2}(x-6)}{\cancel{2} \cdot 2(x+5)} \nonumber

Simplify.

\dfrac{x-6}{2(x+5)} \nonumber

Notice that there are no more factors common to the numerator and denominator.

Try It \PageIndex{13}

Simplify the complex rational expression by using the LCD: \dfrac{\dfrac{3}{x+2}}{\dfrac{5}{x-2}-\dfrac{3}{x^{2}-4}} \nonumber

Answer

\dfrac{3(x-2)}{5 x+7}

Try It \PageIndex{14}

Simplify the complex rational expression by using the LCD: \dfrac{\dfrac{2}{x-7}-\dfrac{1}{x+7}}{\dfrac{6}{x+7}-\dfrac{1}{x^{2}-49}} \nonumber

Answer

\dfrac{x+21}{6 x-43}

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

Example \PageIndex{8}

Simplify the complex rational expression by using the LCD: \dfrac{\dfrac{4}{m^{2}-7 m+12}}{\dfrac{3}{m-3}-\dfrac{2}{m-4}} \nonumber

Solution

Find the LCD of all fractions in the complex rational expression.

The LCD is (m−3)(m−4).

Multiply the numerator and denominator by the LCD.

\dfrac{(m-3)(m-4) \dfrac{4}{(m-3)(m-4)}}{(m-3)(m-4)\left(\dfrac{3}{m-3}-\dfrac{2}{m-4}\right)} \nonumber

Simplify.

\dfrac{\cancel {(m-3)(m-4)}\dfrac{4}{\cancel {(m-3)(m-4)}}}{\cancel {(m-3)}(m-4)\left(\dfrac{3}{\cancel {m-3}}\right)-(m-3)\cancel {(m-4)}\left(\dfrac{2}{\cancel {m-4}}\right)} \nonumber

Simplify.

\dfrac{4}{3(m-4)-2(m-3)} \nonumber

Distribute.

\dfrac{4}{3m-12-2m+6} \nonumber

Combine like terms.

\dfrac{4}{m-6} \nonumber

Try It \PageIndex{15}

Simplify the complex rational expression by using the LCD: \dfrac{\dfrac{3}{x^{2}+7 x+10}}{\dfrac{4}{x+2}+\dfrac{1}{x+5}} \nonumber

Answer

\dfrac{3}{5 x+22}

Try It \PageIndex{16}

Simplify the complex rational expression by using the LCD: \dfrac{\dfrac{4 y}{y+5}+\dfrac{2}{y+6}}{\dfrac{3 y}{y^{2}+11 y+30}} \nonumber

Answer

\dfrac{2\left(2 y^{2}+13 y+5\right)}{3 y}

Example \PageIndex{9}

Simplify the complex rational expression by using the LCD: \dfrac{\dfrac{y}{y+1}}{1+\dfrac{1}{y-1}} \nonumber

Solution

Find the LCD of all fractions in the complex rational expression.

The LCD is (y+1)(y−1).

Multiply the numerator and denominator by the LCD.

\dfrac{(y+1)(y-1) \dfrac{y}{y+1}}{(y+1)(y-1)\left(1+\dfrac{1}{y-1}\right)} \nonumber

Distribute in the denominator and simplify.

\dfrac{\cancel{(y+1)}(y-1) \dfrac{y}{\cancel {y+1}}}{(y+1)(y-1)(1)+(y+1)\cancel{(y-1)}\left(\dfrac{1}{\cancel{(y-1)}}\right)} \nonumber

Simplify.

\dfrac{(y-1) y}{(y+1)(y-1)+(y+1)} \nonumber

Simplify the denominator and leave the numerator factored.

\dfrac{y(y-1)}{y^{2}-1+y+1} \nonumber

\dfrac{y(y-1)}{y^{2}+y} \nonumber

Factor the denominator and remove factors common with the numerator.

\dfrac{\cancel {y}(y-1)}{\cancel {y}(y+1)} \nonumber

Simplify.

\dfrac{y-1}{y+1} \nonumber

Try It \PageIndex{17}

Simplify the complex rational expression by using the LCD: \dfrac{\dfrac{x}{x+3}}{1+\dfrac{1}{x+3}} \nonumber

Answer

\dfrac{x}{x+4}

Try It \PageIndex{18}

Simplify the complex rational expression by using the LCD: \dfrac{1+\dfrac{1}{x-1}}{\dfrac{3}{x+1}} \nonumber

Answer

\dfrac{x(x+1)}{3(x-1)}

Access this online resource for additional instruction and practice with complex fractions.

  • Complex Fractions

Key Concepts

  • How to 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.
  • How to simplify a complex rational expression by using the LCD.
    1. Find the LCD of all fractions in the complex rational expression.
    2. Multiply the numerator and denominator by the LCD.
    3. Simplify the expression.

Glossary

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

This page titled 15.4: Simplify Complex Rational Expressions is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

Support Center

How can we help?