Simplify Complex Rational Expressions

By the end of this section, you will be able to: simplify a complex rational expression by writing it as division, and simplify a complex rational expression by using the LCD.

Simplify a complex rational expression by writing it as division

Complex fractions are fractions in which the numerator or denominator contains a fraction. 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:

\begin{array}{ccc} \cfrac{\tfrac{4}{y-3}}{\tfrac{8}{y^2-9}} & \cfrac{\tfrac{1}{x}+\tfrac{1}{y}}{\tfrac{x}{y}-\tfrac{y}{x}} & \cfrac{\tfrac{2}{x+6}}{\tfrac{4}{x-6}-\tfrac{4}{x^2-36}} \end{array}

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 a complex rational expression earlier in this chapter:

\cfrac{\tfrac{6x^2-7x+2}{4x-8}}{\tfrac{2x^2-8x+3}{x^2-5x+6}}.

Fraction bars tell us to divide, so we rewrite it as a division problem and multiply the first rational expression by the reciprocal of the second. This is one method for simplifying complex rational expressions. First make sure the expression has one fraction over one fraction, and then write it as though you were dividing two fractions.

Example. Simplify the complex rational expression by writing it as division:

\cfrac{\tfrac{6}{x-4}}{\tfrac{3}{x^2-16}}.

Rewrite the complex fraction as division, multiply by the reciprocal, factor, and remove common factors:

\begin{array}{lrcl} \text{Rewrite as division.} & \tfrac{6}{x-4} &\div& \tfrac{3}{x^2-16} \\[10pt] \text{Multiply by the reciprocal.} & \tfrac{6}{x-4} &\cdot& \tfrac{x^2-16}{3} \\[10pt] \text{Factor.} &&& \tfrac{3\cdot2\cdot(x-4)(x+4)}{3(x-4)} \\[10pt] \text{Remove common factors and simplify.} &&& 2(x+4) \end{array}

The original expression has denominators $x-4$ and $x^2-16$ , so it is undefined if $x=4$ or $x=-4$ .

Simplify $\cfrac{\tfrac{2}{x^2-1}}{\tfrac{3}{x+1}}$ by writing it as division.

Simplify $\cfrac{\tfrac{1}{x^2-7x+12}}{\tfrac{2}{x-4}}$ by writing it as division.

Fraction bars act as grouping symbols. To follow the Order of Operations, we simplify the numerator and denominator as much as possible before doing the division.

Example. Simplify by writing the complex rational expression as division:

\cfrac{\tfrac{1}{3}+\tfrac{1}{6}}{\tfrac{1}{2}-\tfrac{1}{3}}.

First simplify the numerator and denominator, and then divide:

\begin{array}{lrcl} \text{Find the LCDs and combine.} &&& \cfrac{\tfrac{2}{6}+\tfrac{1}{6}}{\tfrac{3}{6}-\tfrac{2}{6}} \\[10pt] \text{Simplify numerator and denominator.} &&& \cfrac{\tfrac{3}{6}}{\tfrac{1}{6}} \\[10pt] \text{Rewrite as division.} & \tfrac{3}{6} &\div& \tfrac{1}{6} \\[10pt] \text{Multiply by the reciprocal and simplify.} & \tfrac{3}{6}\cdot\tfrac{6}{1} &=& 3 \end{array}

Simplify $\cfrac{\tfrac{1}{2}+\tfrac{2}{3}}{\tfrac{5}{6}+\tfrac{1}{12}}$ by writing it as division.

Simplify $\cfrac{\tfrac{3}{4}-\tfrac{1}{3}}{\tfrac{1}{8}+\tfrac{5}{6}}$ by writing it as division.

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

Example. Simplify by writing the complex rational expression as division:

\cfrac{\tfrac{1}{x}+\tfrac{1}{y}}{\tfrac{x}{y}-\tfrac{y}{x}}.

Simplify the sum in the numerator and the difference in the denominator:

\begin{array}{lrcl} \text{Combine the numerator.} &&& \tfrac{y+x}{xy} \\[10pt] \text{Combine the denominator.} &&& \tfrac{x^2-y^2}{xy} \\[10pt] \text{Rewrite as division.} & \tfrac{y+x}{xy} &\div& \tfrac{x^2-y^2}{xy} \\[10pt] \text{Multiply by the reciprocal and factor.} &&& \tfrac{y+x}{xy}\cdot\tfrac{xy}{(x-y)(x+y)} \\[10pt] \text{Remove common factors and simplify.} &&& \tfrac{1}{x-y} \end{array}

Simplify $\cfrac{\tfrac{1}{x}+\tfrac{1}{y}}{\tfrac{1}{x}-\tfrac{1}{y}}$ by writing it as division.

Simplify $\cfrac{\tfrac{1}{a}+\tfrac{1}{b}}{\tfrac{1}{a^2}-\tfrac{1}{b^2}}$ by writing it as division.

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. Simplify by writing the complex rational expression as division:

\cfrac{n-\tfrac{4n}{n+5}}{\tfrac{1}{n+5}+\tfrac{1}{n-5}}.

Find common denominators for the numerator and denominator, simplify each, then divide:

\begin{array}{lrcl} \text{Simplify the numerator.} &&& \tfrac{n^2+n}{n+5} \\[10pt] \text{Simplify the denominator.} &&& \tfrac{2n}{(n+5)(n-5)} \\[10pt] \text{Rewrite as division.} & \tfrac{n^2+n}{n+5} &\div& \tfrac{2n}{(n+5)(n-5)} \\[10pt] \text{Multiply by the reciprocal.} &&& \tfrac{n^2+n}{n+5}\cdot\tfrac{(n+5)(n-5)}{2n} \\[10pt] \text{Factor and remove common factors.} &&& \tfrac{n(n+1)(n+5)(n-5)}{(n+5)2n} \\[10pt] \text{Simplify.} &&& \tfrac{(n+1)(n-5)}{2} \end{array}

Simplify $\cfrac{b-\tfrac{3b}{b+5}}{\tfrac{2}{b+5}+\tfrac{1}{b-5}}$ by writing it as division.

Simplify $\cfrac{1-\tfrac{3}{c+4}}{\tfrac{1}{c+4}+\tfrac{c}{3}}$ by writing it as division.

Simplify a complex rational expression by using the LCD

When we solved equations with fractions, we “cleared” the fractions by multiplying by the LCD. We can use that strategy here. We multiply the numerator and denominator by the LCD of all the rational expressions. Since we multiply by $\tfrac{\text{LCD}}{\text{LCD}}$ , we multiply by $1$ , so the value stays the same.

Example. Simplify by using the LCD:

\cfrac{\tfrac{1}{3}+\tfrac{1}{6}}{\tfrac{1}{2}-\tfrac{1}{3}}.

The LCD of all fractions is $6$ . Multiply both the numerator and denominator by $6$ , distribute, and simplify:

\begin{array}{lrcl} \text{Multiply by the LCD.} &&& \cfrac{6\left(\tfrac{1}{3}+\tfrac{1}{6}\right)}{6\left(\tfrac{1}{2}-\tfrac{1}{3}\right)} \\[10pt] \text{Distribute.} &&& \cfrac{6\cdot\tfrac{1}{3}+6\cdot\tfrac{1}{6}}{6\cdot\tfrac{1}{2}-6\cdot\tfrac{1}{3}} \\[10pt] \text{Simplify.} &&& \tfrac{2+1}{3-2}=3 \end{array}

Simplify $\cfrac{\tfrac{1}{2}+\tfrac{1}{5}}{\tfrac{1}{10}+\tfrac{1}{5}}$ by using the LCD.

Simplify $\cfrac{\tfrac{1}{4}+\tfrac{3}{8}}{\tfrac{1}{2}-\tfrac{5}{16}}$ by using the LCD.

Example. Simplify by using the LCD:

\cfrac{\tfrac{1}{x}+\tfrac{1}{y}}{\tfrac{x}{y}-\tfrac{y}{x}}.

The LCD is $xy$ :

\begin{array}{lrcl} \text{Multiply numerator and denominator by }xy. &&& \cfrac{xy\left(\tfrac{1}{x}+\tfrac{1}{y}\right)}{xy\left(\tfrac{x}{y}-\tfrac{y}{x}\right)} \\[10pt] \text{Distribute and simplify.} &&& \tfrac{y+x}{x^2-y^2} \\[10pt] \text{Factor and remove common factors.} &&& \tfrac{y+x}{(x-y)(x+y)} \\[10pt] \text{Simplify.} &&& \tfrac{1}{x-y} \end{array}

Simplify $\cfrac{\tfrac{1}{a}+\tfrac{1}{b}}{\tfrac{a}{b}+\tfrac{b}{a}}$ by using the LCD.

Simplify $\cfrac{\tfrac{1}{x^2}-\tfrac{1}{y^2}}{\tfrac{1}{x}+\tfrac{1}{y}}$ by using the LCD.

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. Simplify by using the LCD:

\cfrac{\tfrac{2}{x+6}}{\tfrac{4}{x-6}-\tfrac{4}{x^2-36}}.

The LCD is $x^2-36=(x+6)(x-6)$ . Multiply numerator and denominator by it:

\begin{array}{lrcl} \text{Multiply by the LCD.} &&& \cfrac{(x+6)(x-6)\tfrac{2}{x+6}}{(x+6)(x-6)\left(\tfrac{4}{x-6}-\tfrac{4}{(x+6)(x-6)}\right)} \\[10pt] \text{Distribute and simplify.} &&& \tfrac{2(x-6)}{4(x+6)-4} \\[10pt] \text{Combine terms and factor.} &&& \tfrac{2(x-6)}{4x+20}=\tfrac{2(x-6)}{4(x+5)} \\[10pt] \text{Remove common factors.} &&& \tfrac{x-6}{2(x+5)} \end{array}

Simplify $\cfrac{\tfrac{3}{x+2}}{\tfrac{5}{x-2}-\tfrac{3}{x^2-4}}$ by using the LCD.

Simplify $\cfrac{\tfrac{2}{x-7}-\tfrac{1}{x+7}}{\tfrac{6}{x+7}-\tfrac{1}{x^2-49}}$ by using the LCD.

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

Example. Simplify by using the LCD:

\cfrac{\tfrac{4}{m^2-7m+12}}{\tfrac{3}{m-3}-\tfrac{2}{m-4}}.

Factor $m^2-7m+12=(m-3)(m-4)$ , so the LCD is $(m-3)(m-4)$ :

\begin{array}{lrcl} \text{Multiply by the LCD.} &&& \cfrac{(m-3)(m-4)\tfrac{4}{(m-3)(m-4)}}{(m-3)(m-4)\left(\tfrac{3}{m-3}-\tfrac{2}{m-4}\right)} \\[10pt] \text{Simplify.} &&& \tfrac{4}{3(m-4)-2(m-3)} \\[10pt] \text{Distribute.} &&& \tfrac{4}{3m-12-2m+6} \\[10pt] \text{Combine like terms.} &&& \tfrac{4}{m-6} \end{array}

Simplify $\cfrac{\tfrac{3}{x^2+7x+10}}{\tfrac{4}{x+2}+\tfrac{1}{x+5}}$ by using the LCD.

Simplify $\cfrac{\tfrac{4y}{y+5}+\tfrac{2}{y+6}}{\tfrac{3y}{y^2+11y+30}}$ by using the LCD.

Example. Simplify by using the LCD:

\cfrac{\tfrac{y}{y+1}}{1+\tfrac{1}{y-1}}.

The LCD is $(y+1)(y-1)$ :

\begin{array}{lrcl} \text{Multiply by the LCD.} &&& \cfrac{(y+1)(y-1)\tfrac{y}{y+1}}{(y+1)(y-1)\left(1+\tfrac{1}{y-1}\right)} \\[10pt] \text{Distribute and simplify.} &&& \tfrac{y(y-1)}{(y+1)(y-1)+(y+1)} \\[10pt] \text{Simplify the denominator.} &&& \tfrac{y(y-1)}{y^2+y} \\[10pt] \text{Factor and remove common factors.} &&& \tfrac{y(y-1)}{y(y+1)} \\[10pt] \text{Simplify.} &&& \tfrac{y-1}{y+1} \end{array}

Simplify $\cfrac{\tfrac{x}{x+3}}{1+\tfrac{1}{x+3}}$ by using the LCD.

Simplify $\cfrac{1+\tfrac{1}{x-1}}{\tfrac{3}{x+1}}$ by using the LCD.

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

Adapted from OpenStax Intermediate Algebra 2e, Section 7.3, by Lynn Marecek and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: reformatted examples and Try It exercises for interactive web use and accessibility.

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