2.15: Simplifying Complex Rational Expressions
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Complex fractions are fractions in which the numerator or denominator contains a fraction. We previously simplified complex fractions like these:
\[\dfrac{\dfrac{3}{4}}{\dfrac{5}{8}} \quad \quad \quad \dfrac{\dfrac{x}{2}}{\dfrac{x y}{6}} \nonumber \]
In this section, we will simplify complex rational expressions, which are rational expressions with rational expressions in the numerator or denominator.
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:
\[\dfrac{\dfrac{4}{y-3}}{\dfrac{8}{y^{2}-9}} \quad \quad \dfrac{\dfrac{1}{x}+\dfrac{1}{y}}{\dfrac{x}{y}-\dfrac{y}{x}} \quad \quad \dfrac{\dfrac{2}{x+6}}{\dfrac{4}{x-6}-\dfrac{4}{x^{2}-36}} \nonumber \]
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 when we looked at dividing rational expressions.
\[\dfrac{\dfrac{6 x^{2}-7 x+2}{4 x-8}}{\dfrac{2 x^{2}-8 x+3}{x^{2}-5 x+6}} \nonumber \]
We noted that fraction bars tell us to divide, so rewrote it as the division problem:
\[\left(\dfrac{6 x^{2}-7 x+2}{4 x-8}\right) \div\left(\dfrac{2 x^{2}-8 x+3}{x^{2}-5 x+6}\right) \nonumber \]
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.
Simplify: \[\dfrac{\dfrac{6}{x-4}}{\dfrac{3}{x^{2}-16}} \nonumber \]
Solution
Rewrite the complex fraction as division. \[\dfrac{6}{x-4} \div \dfrac{3}{x^{2}-16} \nonumber \]
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.
\[\dfrac{6}{x-4} \cdot \dfrac{x^{2}-16}{3} \nonumber \]
Factor.
\[\dfrac{3 \cdot 2}{x-4} \cdot \dfrac{(x-4)(x+4)}{3} \nonumber \]
Multiply.
\[\dfrac{3 \cdot 2(x-4)(x+4)}{3(x-4)}\nonumber \]
Remove common factors.
\[\dfrac{\cancel{3} \cdot 2 \cancel {(x-4)}(x+4)}{\cancel{3} \cancel {(x-4)}} \nonumber \]
Simplify.
\[2(x+4) \nonumber \]
Are there any value(s) 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 \(x=-4\).
Simplify: \[\dfrac{\dfrac{1}{x^{2}-7 x+12}}{\dfrac{2}{x-4}} \nonumber \]
- 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.
Simplify: \[\dfrac{\dfrac{1}{3}+\dfrac{1}{6}}{\dfrac{1}{2}-\dfrac{1}{3}} \nonumber \]
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.
\[3 \nonumber \]
Simplify: \[\dfrac{\dfrac{1}{2}+\dfrac{2}{3}}{\dfrac{5}{6}+\dfrac{1}{12}} \nonumber \]
- Answer
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\(\dfrac{14}{11}\)
We follow the same procedure when the complex rational expression contains variables.
Simplify: \[\dfrac{\dfrac{1}{x}+\dfrac{1}{y}}{\dfrac{x}{y}-\dfrac{y}{x}} \nonumber \]
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.
\[\dfrac{1}{x-y} \nonumber \]
Simplify: \[\dfrac{\dfrac{1}{a}+\dfrac{1}{b}}{\dfrac{1}{a^{2}}-\dfrac{1}{b^{2}}} \nonumber \]
- Answer
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\(\dfrac{a b}{b-a}\)
We summarize the steps here.
- Rewrite the complex rational expression as a division problem.
- Divide the expressions.
Simplify: \[\dfrac{n-\dfrac{4 n}{n+5}}{\dfrac{1}{n+5}+\dfrac{1}{n-5}} \nonumber \]
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.
\[\dfrac{(n+1)(n-5)}{2} \nonumber \]
Simplify: \[\dfrac{1-\dfrac{3}{c+4}}{\dfrac{1}{c+4}+\dfrac{c}{3}} \nonumber \]
- Answer
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\(\dfrac{3}{c+3}\)
Simplifying 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 . 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.
Simplify: \[\dfrac{\dfrac{1}{3}+\dfrac{1}{6}}{\dfrac{1}{2}-\dfrac{1}{3}} \nonumber \]
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.
\[\dfrac{2+1}{3-2} \nonumber \]
\[\dfrac{3}{1}\nonumber \]
\[3\nonumber \]
Simplify: \[\dfrac{\dfrac{1}{4}+\dfrac{3}{8}}{\dfrac{1}{2}-\dfrac{5}{16}} \nonumber \]
- Answer
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\(\dfrac{10}{3}\)
We will use the same example as in . Decide which method works better for you.
Simplify: \[\dfrac{\dfrac{1}{x}+\dfrac{1}{y}}{\dfrac{x}{y}-\dfrac{y}{x}} \nonumber \]
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.
\[\dfrac{1}{x-y} \nonumber \]
Simplify: \[\dfrac{\dfrac{1}{x^{2}}-\dfrac{1}{y^{2}}}{\dfrac{1}{x}+\dfrac{1}{y}} \nonumber \]
- Answer
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\(\dfrac{y-x}{x y}\)
- 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.
Simplify: \[\dfrac{\dfrac{2}{x+6}}{\dfrac{4}{x-6}-\dfrac{4}{x^{2}-36}} \nonumber \]
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.
Simplify: \[\dfrac{\dfrac{2}{x-7}-\dfrac{1}{x+7}}{\dfrac{6}{x+7}-\dfrac{1}{x^{2}-49}} \nonumber \]
- Answer
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\(\dfrac{x+21}{6 x-43}\)
Be sure to factor the denominators first. Proceed carefully and write all steps to help avoid errors.
Simplify: \[\dfrac{\dfrac{4}{m^{2}-7 m+12}}{\dfrac{3}{m-3}-\dfrac{2}{m-4}} \nonumber \]
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.
\[\dfrac{4}{m-6} \nonumber \]
Simplify: \[\dfrac{\dfrac{4 y}{y+5}+\dfrac{2}{y+6}}{\dfrac{3 y}{y^{2}+11 y+30}} \nonumber \]
- Answer
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\(\dfrac{2\left(2 y^{2}+13 y+5\right)}{3 y}\)
Simplify: \[\dfrac{\dfrac{y}{y+1}}{1+\dfrac{1}{y-1}} \nonumber \]
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.
\[\dfrac{y-1}{y+1} \nonumber \]
Simplify: \[\dfrac{\dfrac{x}{x+3}}{1+\dfrac{1}{x+3}} \nonumber \]
- Answer
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\(\dfrac{x}{x+4}\)