15.4: Simplify Complex Rational Expressions
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- Aug 13, 2020
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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.
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:
4y−38y2−91x+1yxy−yx2x+64x−6−4x2−36
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.
6x2−7x+24x−82x2−8x+3x2−5x+6
We noted that fraction bars tell us to divide, so rewrote it as the division problem:
(6x2−7x+24x−8)÷(2x2−8x+3x2−5x+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 15.4.1
Simplify the complex rational expression by writing it as division: 6x−43x2−16
Solution
Rewrite the complex fraction as division. 6x−4÷3x2−16
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.
6x−4⋅x2−163
Factor.
3⋅2x−4⋅(x−4)(x+4)3
Multiply.
3⋅2(x−4)(x+4)3(x−4)
Remove common factors.
3⋅2(x−4)(x+4)3(x−4)
Simplify.
2(x+4)
Are there any value(s) of x that should not be allowed? The original complex rational expression had denominators of x−4 and x2−16 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
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\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.
- Rewrite the complex rational expression as a division problem.
- 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.
- 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 \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.
- Simplify the numerator and denominator.
- Rewrite the complex rational expression as a division problem.
- Divide the expressions.
- 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.
Glossary
- complex rational expression
- A complex rational expression is a rational expression in which the numerator and/or denominator contains a rational expression.