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Intermediate Algebra
Rational Expressions and Functions
Simplify Complex Rational Expressions

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
  • 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:

4y−38y2−91x+1yxy−yx2x+64x−6−4x2−36 \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:

6x2−7x+24x−82x2−8x+3x2−5x+6. \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:

6x−43x2−16. \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:

Rewrite as division.6x−4÷3x2−16Multiply by the reciprocal.6x−4⋅x2−163Factor.3⋅2⋅(x−4)(x+4)3(x−4)Remove common factors and simplify.2(x+4) \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−4x-4 and x2−16x^2-16, so it is undefined if x=4x=4 or x=−4x=-4.

Simplify2x2−13x+1\cfrac{\tfrac{2}{x^2-1}}{\tfrac{3}{x+1}}by writing it as division.

Simplify1x2−7x+122x−4\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:

13+1612−13. \cfrac{\tfrac{1}{3}+\tfrac{1}{6}}{\tfrac{1}{2}-\tfrac{1}{3}}.

First simplify the numerator and denominator, and then divide:

Find the LCDs and combine.26+1636−26Simplify numerator and denominator.3616Rewrite as division.36÷16Multiply by the reciprocal and simplify.36⋅61=3 \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}

Simplify12+2356+112\cfrac{\tfrac{1}{2}+\tfrac{2}{3}}{\tfrac{5}{6}+\tfrac{1}{12}}by writing it as division.

Simplify34−1318+56\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:

1x+1yxy−yx. \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:

Combine the numerator.y+xxyCombine the denominator.x2−y2xyRewrite as division.y+xxy÷x2−y2xyMultiply by the reciprocal and factor.y+xxy⋅xy(x−y)(x+y)Remove common factors and simplify.1x−y \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}

Simplify1x+1y1x−1y\cfrac{\tfrac{1}{x}+\tfrac{1}{y}}{\tfrac{1}{x}-\tfrac{1}{y}}by writing it as division.

Simplify1a+1b1a2−1b2\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.

  1. Simplify the numerator and denominator.
  2. Rewrite the complex rational expression as a division problem.
  3. Divide the expressions.

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

n−4nn+51n+5+1n−5. \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:

Simplify the numerator.n2+nn+5Simplify the denominator.2n(n+5)(n−5)Rewrite as division.n2+nn+5÷2n(n+5)(n−5)Multiply by the reciprocal.n2+nn+5⋅(n+5)(n−5)2nFactor and remove common factors.n(n+1)(n+5)(n−5)(n+5)2nSimplify.(n+1)(n−5)2 \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}

Simplifyb−3bb+52b+5+1b−5\cfrac{b-\tfrac{3b}{b+5}}{\tfrac{2}{b+5}+\tfrac{1}{b-5}}by writing it as division.

Simplify1−3c+41c+4+c3\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 LCDLCD\tfrac{\text{LCD}}{\text{LCD}}, we multiply by 11, so the value stays the same.

Example. Simplify by using the LCD:

13+1612−13. \cfrac{\tfrac{1}{3}+\tfrac{1}{6}}{\tfrac{1}{2}-\tfrac{1}{3}}.

The LCD of all fractions is 66. Multiply both the numerator and denominator by 66, distribute, and simplify:

Multiply by the LCD.6(13+16)6(12−13)Distribute.6⋅13+6⋅166⋅12−6⋅13Simplify.2+13−2=3 \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}

Simplify12+15110+15\cfrac{\tfrac{1}{2}+\tfrac{1}{5}}{\tfrac{1}{10}+\tfrac{1}{5}}by using the LCD.

Simplify14+3812−516\cfrac{\tfrac{1}{4}+\tfrac{3}{8}}{\tfrac{1}{2}-\tfrac{5}{16}}by using the LCD.

Example. Simplify by using the LCD:

1x+1yxy−yx. \cfrac{\tfrac{1}{x}+\tfrac{1}{y}}{\tfrac{x}{y}-\tfrac{y}{x}}.

The LCD is xyxy:

Multiply numerator and denominator by xy.xy(1x+1y)xy(xy−yx)Distribute and simplify.y+xx2−y2Factor and remove common factors.y+x(x−y)(x+y)Simplify.1x−y \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}

Simplify1a+1bab+ba\cfrac{\tfrac{1}{a}+\tfrac{1}{b}}{\tfrac{a}{b}+\tfrac{b}{a}}by using the LCD.

Simplify1x2−1y21x+1y\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.

  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.

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

Example. Simplify by using the LCD:

2x+64x−6−4x2−36. \cfrac{\tfrac{2}{x+6}}{\tfrac{4}{x-6}-\tfrac{4}{x^2-36}}.

The LCD is x2−36=(x+6)(x−6)x^2-36=(x+6)(x-6). Multiply numerator and denominator by it:

Multiply by the LCD.(x+6)(x−6)2x+6(x+6)(x−6)(4x−6−4(x+6)(x−6))Distribute and simplify.2(x−6)4(x+6)−4Combine terms and factor.2(x−6)4x+20=2(x−6)4(x+5)Remove common factors.x−62(x+5) \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}

Simplify3x+25x−2−3x2−4\cfrac{\tfrac{3}{x+2}}{\tfrac{5}{x-2}-\tfrac{3}{x^2-4}}by using the LCD.

Simplify2x−7−1x+76x+7−1x2−49\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:

4m2−7m+123m−3−2m−4. \cfrac{\tfrac{4}{m^2-7m+12}}{\tfrac{3}{m-3}-\tfrac{2}{m-4}}.

Factor m2−7m+12=(m−3)(m−4)m^2-7m+12=(m-3)(m-4), so the LCD is (m−3)(m−4)(m-3)(m-4):

Multiply by the LCD.(m−3)(m−4)4(m−3)(m−4)(m−3)(m−4)(3m−3−2m−4)Simplify.43(m−4)−2(m−3)Distribute.43m−12−2m+6Combine like terms.4m−6 \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}

Simplify3x2+7x+104x+2+1x+5\cfrac{\tfrac{3}{x^2+7x+10}}{\tfrac{4}{x+2}+\tfrac{1}{x+5}}by using the LCD.

Simplify4yy+5+2y+63yy2+11y+30\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:

yy+11+1y−1. \cfrac{\tfrac{y}{y+1}}{1+\tfrac{1}{y-1}}.

The LCD is (y+1)(y−1)(y+1)(y-1):

Multiply by the LCD.(y+1)(y−1)yy+1(y+1)(y−1)(1+1y−1)Distribute and simplify.y(y−1)(y+1)(y−1)+(y+1)Simplify the denominator.y(y−1)y2+yFactor and remove common factors.y(y−1)y(y+1)Simplify.y−1y+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}

Simplifyxx+31+1x+3\cfrac{\tfrac{x}{x+3}}{1+\tfrac{1}{x+3}}by using the LCD.

Simplify1+1x−13x+1\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.

Practice

Simplify a complex rational expression by writing it as division

Simplify2aa+44a2a2−16\cfrac{\tfrac{2a}{a+4}}{\tfrac{4a^2}{a^2-16}}by writing it as division.

Simplifyx−2xx+31x+3+1x−3\cfrac{x-\tfrac{2x}{x+3}}{\tfrac{1}{x+3}+\tfrac{1}{x-3}}by writing it as division.

Simplify a complex rational expression by using the LCD

Simplify2x+53x−5+1x2−25\cfrac{\tfrac{2}{x+5}}{\tfrac{3}{x-5}+\tfrac{1}{x^2-25}}by using the LCD.

Simplify4a2−2a−151a−5+2a+3\cfrac{\tfrac{4}{a^2-2a-15}}{\tfrac{1}{a-5}+\tfrac{2}{a+3}}by using the LCD.

Simplify2+1p−35p−3\cfrac{2+\tfrac{1}{p-3}}{\tfrac{5}{p-3}}by using the LCD.

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, and adapted selected end-of-section exercises into an interactive Practice block.

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