Simplify Complex Rational Expressions

Complex fractions are fractions in which the numerator or denominator contains a fraction. In Chapter 1 we simplified complex fractions like these:

\frac{\frac{3}{4}}{\frac{5}{8}} \frac{\frac{x}{2}}{\frac{x y}{6}}

In this section we will simplify complex rational expressions, which are rational expressions with rational expressions in the numerator or denominator.

Here are a few complex rational expressions:

\frac{\frac{4}{y - 3}}{\frac{8}{y^{2} - 9}} \frac{\frac{1}{x} + \frac{1}{y}}{\frac{x}{y} - \frac{y}{x}} \frac{\frac{2}{x + 6}}{\frac{4}{x - 6} - \frac{4}{x^{2} - 36}}

Remember, we always exclude values that would make any denominator zero.

We will use two methods to simplify complex rational expressions.

Simplify a Complex Rational Expression by Writing it as Division

We have already seen this complex rational expression earlier in this chapter.

\frac{\frac{6 x^{2} - 7 x + 2}{4 x - 8}}{\frac{2 x^{2} - 8 x + 3}{x^{2} - 5 x + 6}}

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

(\frac{6 x^{2} - 7 x + 2}{4 x - 8}) \div (\frac{2 x^{2} - 8 x + 3}{x^{2} - 5 x + 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 rational expressions. We write it as if we were dividing two fractions.

Simplify: $\frac{\frac{4}{y - 3}}{\frac{8}{y^{2} - 9}} .$

Solution

Step-by-step process for simplifying a complex rational expression into its most reduced form.
	$\frac{\frac{4}{y - 3}}{\frac{8}{y^{2} - 9}}$
Rewrite the complex fraction as division.	$\frac{4}{y - 3} \div \frac{8}{y^{2} - 9}$
Rewrite as the product of first times the reciprocal of the second.	$\frac{4}{y - 3} \cdot \frac{y^{2} - 9}{8}$
Multiply.	$\frac{4 (y^{2} - 9)}{8 (y - 3)}$
Factor to look for common factors.	$\frac{4 (y - 3) (y + 3)}{4 \cdot 2 (y - 3)}$
Remove common factors.	$\frac{4 (y - 3) (y + 3)}{4 \cdot 2 (y - 3)}$
Simplify.	$\frac{y + 3}{2}$

Are there any value(s) of $y$ that should not be allowed? The simplified rational expression has just a constant in the denominator. But the original complex rational expression had denominators of $y - 3$ and $y^{2} - 9$ . This expression would be undefined if $y = 3$ or $y = −3$ .

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: $\frac{\frac{1}{3} + \frac{1}{6}}{\frac{1}{2} - \frac{1}{3}} .$

Solution

	$A mathematical expression featuring a complex fraction. The numerator shows the sum of one-third and one-sixth, while the denominator displays the difference between one-half and one-third.$
Simplify the numerator and denominator.
Find the LCD and add the fractions in the numerator. Find the LCD and add the fractions in the denominator.	$A complex fraction showing addition and subtraction of fractions. Red numbers highlight the multiplication steps to find common denominators in both the numerator and denominator.$
Simplify the numerator and denominator.	$A complex fraction mathematical problem showing (2/6 + 1/6) divided by (3/6 - 2/6).$
Simplify the numerator and denominator, again.	$A mathematical expression showing the division of two fractions, (3/6) by (1/6), which simplifies to 3. This illustrates how to divide fractions with a common denominator.$
Rewrite the complex rational expression as a division problem.	$A mathematical expression showing the division of two fractions: three-sixths divided by one-sixth.$
Multiply the first times by the reciprocal of the second.	$A mathematical expression showing the multiplication of two fractions: 3/6 multiplied by 6/1.$
Simplify.

How to Simplify a Complex Rational Expression by Writing it as Division

Simplify: $\frac{\frac{1}{x} + \frac{1}{y}}{\frac{x}{y} - \frac{y}{x}} .$

Solution

$The above image has three columns. The image shows steps on how to divide complex rational expressions in three steps. Step one is to simplify the numerator and denominator. We will simplify the sum in the numerator and difference in the denominator for the example 1 divided by x plus 1 divided by y divided by x divided by y minus y divided by x. Find a common denominator and add the fractions in the numerator and find a common denominator and subtract the fractions in the numerator to get 1 times y divided by x times y plus 1 times x divided by y times x divided by x times x divided by y times x minus y times y divided by x times y. Then, we get y divided by x y plus x plus x y divided by x squared divided by x y minus y squared divided by x y. We now have just one rational expression in the numerator and one in the denominator, y plus x divided by x y divided by x squared minus y squared divided by x y.$ Step two is to rewrite the complex rational expression as a division problem. We write the numerator divided by the denominator.

Step two is to rewrite the complex rational expression as a division problem. We write the numerator divided by the denominator.

Step three is to divide the expressions. Multiply the first by the reciprocal of the second to get y plus x divided by x y times x y divided by x squared minus y squared. Factor any expressions if possible. We now have x y times y plus x divided by x y times x minus y times x plus y. Remove common factors. Cross out x, y and y plus x from the numerator. Cross out x, y and x plus y from the denominator. Simplify to get 1 divided by x minus y.

Simplify: $\frac{n - \frac{4 n}{n + 5}}{\frac{1}{n + 5} + \frac{1}{n - 5}} .$

Solution

	$A complex algebraic fraction involving variables 'n' and constants, with a numerator of (n - 4n/(n+5)) and a denominator of (1/(n+5) + 1/(n-5)).$
Simplify the numerator and denominator.
Find the LCD and add the fractions in the numerator. Find the LCD and add the fractions in the denominator.	$An algebraic expression presented as a complex fraction. The numerator involves the subtraction of two rational terms, and the denominator involves the addition of two rational terms, all featuring the variable 'n'.$
Simplify the numerators.	$A complex algebraic fraction involving the variable 'n'. The numerator is the subtraction of (n^2+5n)/(n+5) and 4n/(n+5), all divided by the sum of (n-5)/((n+5)(n-5)) and (n+5)/((n-5)(n+5)).$
Subtract the rational expressions in the numerator and add in the denominator. Simplify.	$A complex fraction with algebraic expressions, where the numerator is (n^2 + n)/(n + 5) and the denominator is 2n/((n + 5)(n - 5)).$
Rewrite as fraction division.	$The image displays a mathematical expression: the fraction (n^2 + n) / (n + 5) divided by the fraction 2n / ((n + 5)(n - 5)).$
Multiply the first times the reciprocal of the second.	$An algebraic expression showing the product of two fractions. The first fraction is (n^2 + n) divided by (n + 5). The second fraction is ((n + 5)(n - 5)) divided by (2n).$
Factor any expressions if possible.	$A mathematical expression showing a fraction with numerator n(n+1)(n+5)(n-5) and denominator (n+5)2n.$
Remove common factors.	$A fractional algebraic expression with factors n, (n+1), (n+5), and (n-5) in the numerator, and (n+5) and 2n in the denominator, featuring cancellation marks for (n+5) and n.$
Simplify.	$A mathematical expression showing the fraction (n+1)(n-5) divided by 2. The numerator consists of the product of two binomials, (n+1) and (n-5), and the denominator is the number 2.$

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 LCD of all the rational expressions.

Let’s look at the complex rational expression we simplified one way in Example 2. We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by $\frac{LCD}{LCD}$ we are multiplying by 1, so the value stays the same.

Simplify: $\frac{\frac{1}{3} + \frac{1}{6}}{\frac{1}{2} - \frac{1}{3}} .$

Solution

	$A complex fraction mathematical expression: the sum of one-third and one-sixth, divided by the difference between one-half and one-third. It represents a multi-step fraction calculation.$
The LCD of all the fractions in the whole expression is 6.
Clear the fractions by multiplying the numerator and denominator by that LCD.
Distribute.	$A complex fraction where the numerator is (61/3 + 61/6) and the denominator is (61/2 - 61/3), with the number 6 highlighted in red.$
Simplify.	$A mathematical fraction displays the expression (2 + 1) divided by (3 - 2) in black text on a white background.$
	$The image displays the fraction 3/1, with the number 3 positioned above a horizontal fraction bar and the number 1 directly below the bar, set against a plain white background.$

How to Simplify a Complex Rational Expression by Using the LCD

Simplify: $\frac{\frac{1}{x} + \frac{1}{y}}{\frac{x}{y} - \frac{y}{x}} .$

Solution

$The above image has 3 columns. It shows the steps on how to simplify a complex rational expression using the LCD for 1 divided by x plus 1 divided by y divided by x divided by y minus y divided by x. Step one is to find the LCD of all fractions in the complex rational expression. The LCD of all the fractions is x y. Multiply the numerator and denominator by the LCD.$ Step two is to multiply both the numerator and denominator by x y to get x y times 1 divided by x plus 1 divided by y divided x y times x divided by y minus y divided by x.

Step two is to multiply both the numerator and denominator by x y to get x y times 1 divided by x plus 1 divided by y divided x y times x divided by y minus y divided by x.

Step three is to simplify the expression. Distribute to get x y times 1 divided by x plus x y times 1 divided y divided by x y times x divided by y minus x y times y divided by x. Simplify to get y plus x divided by x squared minus y squared. Remove common factors. Cross out y plus x in the numerator. Cross out x plus y in the numerator. Simplify to get 1 divided by x minus y.

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

Simplify: $\frac{\frac{2}{x + 6}}{\frac{4}{x - 6} - \frac{4}{x^{2} - 36}} .$

Solution

	$A complex algebraic fraction with 2/(x+6) in the numerator and a difference of two fractions, 4/(x-6) - 4/(x^2-36), in the denominator.$
Find the LCD of all fractions in the complex rational expression. The LCD is $(x + 6) (x - 6)$ .
Multiply the numerator and denominator by the LCD.	$A complex algebraic fraction demonstrating a step where both numerator and denominator are multiplied by the common factor (x+6)(x-6), highlighted in red, to simplify the expression.$
Simplify the expression.
Distribute in the denominator.	$A step in simplifying a rational expression, illustrating the multiplication of the numerator and denominator by the common factor (x+6)(x-6) to remove inner fractions.$
Simplify.	$Algebraic expression being simplified, demonstrating cancellation of common factors (x+6) and (x-6) in numerator and denominator to reduce the complex fraction.$
Simplify.	$A mathematical fraction is shown. The numerator is 2(x - 6) and the denominator is 4(x + 6) - 4.$
To simplify the denominator, distribute and combine like terms.	$A mathematical expression shown as a fraction with 2(x-6) in the numerator and 4x+20 in the denominator, set against a plain white background.$
Remove common factors.	$A mathematical expression showing a fraction with 2(x-6) in the numerator and 2(2x+10) in the denominator.$
Simplify.	$A mathematical expression showing the fraction (x-6) over (2x+10).$
Notice that there are no more factors common to the numerator and denominator.

Simplify: $\frac{\frac{4}{m^{2} - 7 m + 12}}{\frac{3}{m - 3} - \frac{2}{m - 4}} .$

Solution

	$A complex algebraic fraction. The numerator is 4, and the denominator is another fraction with m^2 - 7m + 12 as its numerator and (3/(m-3) - 2/(m-4)) as its denominator.$
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.	$A step in simplifying a complex algebraic fraction by multiplying by the common denominator (m-3)(m-4) to clear fractions in both the numerator and denominator.$
Simplify.
Simplify.	$A mathematical expression featuring a fraction with '4' as the numerator and '3(m-4) - 2(m-3)' as the denominator, suitable for algebraic simplification.$
Distribute.	$The image shows a mathematical fraction where the numerator is 4 and the denominator is 3m - 12 - 2m + 6.$
Combine like terms.	$A mathematical expression showing the fraction 4 over (m minus 6).$

Simplify: $\frac{\frac{y}{y + 1}}{1 + \frac{1}{y - 1}} .$

Solution

	$A complex fraction with the numerator as y divided by (y+1) and the denominator as 1 plus 1 divided by (y-1).$
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.	$An algebraic fraction where both numerator and denominator are multiplied by the red terms (y+1)(y-1). The expression involves a fraction y/(y+1) in the numerator and (1 + 1/(y-1)) in the denominator.$
Distribute in the denominator and simplify.
Simplify.	$A mathematical expression showing a fraction. The numerator is (y-1)y. The denominator is (y+1)(y-1) + (y+1).$
Simplify the denominator, and leave the numerator factored.	$A mathematical fraction with y(y-1) in the numerator and y^2 - 1 + y + 1 in the denominator, set against a plain white background.$
	$A mathematical expression showing a fraction with y(y-1) in the numerator and y^2 + y in the denominator, written in black text on a white background.$
Factor the denominator, and remove factors common with the numerator.	$A mathematical expression showing the fraction x(y-1) over x(y+1), where the 'x' terms are crossed out, indicating simplification to (y-1) over (y+1).$
Simplify.	$A mathematical expression showing the fraction (y - 1) / (y + 1).$

Key Concepts

To Simplify a 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.
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.

Practice Makes Perfect

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

$\frac{\frac{2 a}{a + 4}}{\frac{4 a^{2}}{a^{2} - 16}}$

Solution

$\frac{a - 4}{2 a}$

$\frac{\frac{3 b}{b - 5}}{\frac{b^{2}}{b^{2} - 25}}$

$\frac{\frac{5}{c^{2} + 5 c - 14}}{\frac{10}{c + 7}}$

Solution

$\frac{1}{2 (c - 2)}$

$\frac{\frac{8}{d^{2} + 9 d + 18}}{\frac{12}{d + 6}}$

$\frac{\frac{1}{2} + \frac{5}{6}}{\frac{2}{3} + \frac{7}{9}}$

Solution

$\frac{12}{13}$

$\frac{\frac{1}{2} + \frac{3}{4}}{\frac{3}{5} + \frac{7}{10}}$

$\frac{\frac{2}{3} - \frac{1}{9}}{\frac{3}{4} + \frac{5}{6}}$

Solution

$\frac{20}{57}$

$\frac{\frac{1}{2} - \frac{1}{6}}{\frac{2}{3} + \frac{3}{4}}$

$\frac{\frac{n}{m} + \frac{1}{n}}{\frac{1}{n} - \frac{n}{m}}$

Solution

$\frac{n^{2} + m}{m - n^{2}}$

$\frac{\frac{1}{p} + \frac{p}{q}}{\frac{q}{p} - \frac{1}{q}}$

$\frac{\frac{1}{r} + \frac{1}{t}}{\frac{1}{r^{2}} - \frac{1}{t^{2}}}$

Solution

$\frac{r t}{t - r}$

$\frac{\frac{2}{v} + \frac{2}{w}}{\frac{1}{v^{2}} - \frac{1}{w^{2}}}$

$\frac{x - \frac{2 x}{x + 3}}{\frac{1}{x + 3} + \frac{1}{x - 3}}$

Solution

$\frac{(x + 1) (x - 3)}{2}$

$\frac{y - \frac{2 y}{y - 4}}{\frac{2}{y - 4} - \frac{2}{y + 4}}$

$\frac{2 - \frac{2}{a + 3}}{\frac{1}{a + 3} + \frac{a}{2}}$

Solution

$\frac{4}{a + 1}$

$\frac{4 - \frac{4}{b - 5}}{\frac{1}{b - 5} + \frac{b}{4}}$

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

$\frac{\frac{1}{3} + \frac{1}{8}}{\frac{1}{4} + \frac{1}{12}}$

Solution

$\frac{11}{8}$

$\frac{\frac{1}{4} + \frac{1}{9}}{\frac{1}{6} + \frac{1}{12}}$

$\frac{\frac{5}{6} + \frac{2}{9}}{\frac{7}{18} - \frac{1}{3}}$

Solution

$19$

$\frac{\frac{1}{6} + \frac{4}{15}}{\frac{3}{5} - \frac{1}{2}}$

$\frac{\frac{c}{d} + \frac{1}{d}}{\frac{1}{d} - \frac{d}{c}}$

Solution

$\frac{c^{2} + c}{c - d^{2}}$

$\frac{\frac{1}{m} + \frac{m}{n}}{\frac{n}{m} - \frac{1}{n}}$

$\frac{\frac{1}{p} + \frac{1}{q}}{\frac{1}{p^{2}} - \frac{1}{q^{2}}}$

Solution

$\frac{p q}{q - p}$

$\frac{\frac{2}{r} + \frac{2}{t}}{\frac{1}{r^{2}} - \frac{1}{t^{2}}}$

$\frac{\frac{2}{x + 5}}{\frac{3}{x - 5} + \frac{1}{x^{2} - 25}}$

Solution

$\frac{2 x - 10}{3 x + 16}$

$\frac{\frac{5}{y - 4}}{\frac{3}{y + 4} + \frac{2}{y^{2} - 16}}$

$\frac{\frac{5}{z^{2} - 64} + \frac{3}{z + 8}}{\frac{1}{z + 8} + \frac{2}{z - 8}}$

Solution

$\frac{3 z - 19}{3 z + 8}$

$\frac{\frac{3}{s + 6} + \frac{5}{s - 6}}{\frac{1}{s^{2} - 36} + \frac{4}{s + 6}}$

$\frac{\frac{4}{a^{2} - 2 a - 15}}{\frac{1}{a - 5} + \frac{2}{a + 3}}$

Solution

$\frac{4}{3 a - 7}$

$\frac{\frac{5}{b^{2} - 6 b - 27}}{\frac{3}{b - 9} + \frac{1}{b + 3}}$

$\frac{\frac{5}{c + 2} - \frac{3}{c + 7}}{\frac{5 c}{c^{2} + 9 c + 14}}$

Solution

$\frac{2 c + 29}{5 c}$

$\frac{\frac{6}{d - 4} - \frac{2}{d + 7}}{\frac{2 d}{d^{2} + 3 d - 28}}$

$\frac{2 + \frac{1}{p - 3}}{\frac{5}{p - 3}}$

Solution

$\frac{(2 p - 5)}{5}$

$\frac{\frac{n}{n - 2}}{3 + \frac{5}{n - 2}}$

$\frac{\frac{m}{m + 5}}{4 + \frac{1}{m - 5}}$

Solution

$\frac{m (m - 5)}{4 m^{2} + m - 95}$

$\frac{7 + \frac{2}{q - 2}}{\frac{1}{q + 2}}$

Simplify

In the following exercises, use either method.

$\frac{\frac{3}{4} - \frac{2}{7}}{\frac{1}{2} + \frac{5}{14}}$

Solution

$\frac{13}{24}$

$\frac{\frac{v}{w} + \frac{1}{v}}{\frac{1}{v} - \frac{v}{w}}$

$\frac{\frac{2}{a + 4}}{\frac{1}{a^{2} - 16}}$

Solution

$2 (a - 4)$

$\frac{\frac{3}{b^{2} - 3 b - 40}}{\frac{5}{b + 5} - \frac{2}{b - 8}}$

$\frac{\frac{3}{m} + \frac{3}{n}}{\frac{1}{m^{2}} - \frac{1}{n^{2}}}$

Solution

$\frac{3 m n}{n - m}$

$\frac{\frac{2}{r - 9}}{\frac{1}{r + 9} + \frac{3}{r^{2} - 81}}$

$\frac{x - \frac{3 x}{x + 2}}{\frac{3}{x + 2} + \frac{3}{x - 2}}$

Solution

$\frac{(x - 1) (x - 2)}{6}$

$\frac{\frac{y}{y + 3}}{2 + \frac{1}{y - 3}}$

Everyday Math

Electronics The resistance of a circuit formed by connecting two resistors in parallel is $\frac{1}{\frac{1}{R_{1}} + \frac{1}{R_{2}}}$ .

ⓐ Simplify the complex fraction $\frac{1}{\frac{1}{R_{1}} + \frac{1}{R_{2}}}$ .
ⓑ Find the resistance of the circuit when $R_{1} = 8$ and $R_{2} = 12$ .

Solution

ⓐ $\frac{R_{1} R_{2}}{R_{2} + R_{1}}$ ⓑ $\frac{24}{5}$

Ironing Lenore can do the ironing for her family’s business in $h$ hours. Her daughter would take $h + 2$ hours to get the ironing done. If Lenore and her daughter work together, using 2 irons, the number of hours it would take them to do all the ironing is $\frac{1}{\frac{1}{h} + \frac{1}{h + 2}}$ .

ⓐ Simplify the complex fraction $\frac{1}{\frac{1}{h} + \frac{1}{h + 2}}$ .
ⓑ Find the number of hours it would take Lenore and her daughter, working together, to get the ironing done if $h = 4$ .

Writing Exercises

In this section, you learned to simplify the complex fraction $\frac{\frac{3}{x + 2}}{\frac{x}{x^{2} - 4}}$ two ways:

rewriting it as a division problem

multiplying the numerator and denominator by the LCD

Which method do you prefer? Why?

Solution

Answers will vary.

Efraim wants to start simplifying the complex fraction $\frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a} - \frac{1}{b}}$ by cancelling the variables from the numerator and denominator. Explain what is wrong with Efraim’s plan.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

The above image is four columns and three rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “simplify a complex rational expression by writing it as division.”, the next row reads “simplify a complex rational expression by using the LCD.” The remaining columns are blank.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

complex rational expression

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