Simplify Complex Rational Expressions

Simplify a Complex Rational Expression by Writing it as Division

Complex fractions are fractions in which the numerator or denominator contains a fraction. We previously simplified complex fractions like these:

\begin{matrix} \frac{\frac{3}{4}}{\frac{5}{8}} & \frac{\frac{x}{2}}{\frac{x y}{6}} \end{matrix}

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.

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 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 the complex rational expression by writing it as division: $\frac{\frac{6}{x - 4}}{\frac{3}{x^{2} - 16}} .$

Solution

Step-by-step simplification of a complex rational expression involving algebraic terms.
	$\frac{\frac{6}{x - 4}}{\frac{3}{x^{2} - 16}}$
Rewrite the complex fraction as division.	$\frac{6}{x - 4} \div \frac{3}{x^{2} - 16}$
Rewrite as the product of first times the reciprocal of the second.	$\frac{6}{x - 4} \cdot \frac{x^{2} - 16}{3}$
Factor.	$\frac{3 \cdot 2}{x - 4} \cdot \frac{(x - 4) (x + 4)}{3}$
Multiply.	$\frac{3 \cdot 2 (x - 4) (x + 4)}{3 (x - 4)}$
Remove common factors.	$\frac{3 \cdot 2 (x - 4) (x + 4)}{3 (x - 4)}$
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 $x^{2} - 16 .$ This expression would be undefined if $x = 4$ or $x = −4 .$

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 the complex rational expression by writing it as division: $\frac{\frac{1}{3} + \frac{1}{6}}{\frac{1}{2} - \frac{1}{3}} .$

Solution

	$A fraction with the sum of 1/3 and 1/6 in the numerator, and the difference of 1/2 and 1/3 in the denominator, representing a complex mathematical expression.$
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.	$A complex fraction illustrating the addition of 1/3 and 1/6 in the numerator, and the subtraction of 1/3 from 1/2 in the denominator, with common denominator steps shown.$
Simplify the numerator and denominator.	$A fraction where the numerator is 2/6 + 1/6 and the denominator is 3/6 - 2/6, demonstrating addition and subtraction of fractions with a common denominator.$
Rewrite the complex rational expression as a division problem.	$A mathematical expression showing the division of two fractions: 3/6 ÷ 1/6.$
Multiply the first by the reciprocal of the second.	$A mathematical expression showing the multiplication of two fractions: 3/6 multiplied by 6/1. The 'dot' symbol is used to denote multiplication between the fractions.$
Simplify.	3

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

How to Simplify a Complex Rational Expression using Division

Simplify the complex rational expression by writing it as division: $\frac{\frac{1}{x} + \frac{1}{y}}{\frac{x}{y} - \frac{y}{x}} .$

Solution

$Step 1 is to simplify the sum in the numerator and the difference in the denominator of complex rational expression, the quantity 1 divided by x plus 1 divided by y all divided by the quantity x divided by y minus y divided by x. The common denominator of the fractions in the complex rational expression is x y. Multiply the numerator and denominator of 1 divided by x by y over y. Multiply the numerator and denominator of 1 divided by y by x over x. Multiply the numerator and denominator of x divided by y by x over x. Multiply the numerator and denominator of y over x by y over y. The result is the quantity y divided by x y plus x divided by x y all divided by the quantity x squared divided by x y minus y squared divided by x y. Add the fractions in the numerator and subtract the fractions in the denominator. The result is the sum of y and x divided by x y all divided by the difference between x squared and y squared divided by x y. We now have just one rational expression in the numerator and one in the denominator.$

Step 2 is to rewrite the complex rational expression as a division problem. Write the numerator divided by the denominator. The result is the quantity of the sum y and x divided by x y all divided by the quantity of the difference between x squared and y squared divided by x y.

Step 3 is to divided the expressions. Multiply the first expression by the reciprocal of the second expression. The result is the quantity of the sum y and x divided by x y times the quantity x y divided by the difference between x squared and y squared. Factor any expressions if possible. The result is the product of x y and the sum of y and x all divided by the product of x y, the difference between x and y, and the sum of x and y. Remove the common factors, x y and the sum of x and y. Simplify. The result is 1 divided by the quantity x minus y.

We summarize the steps here.

Simplify the complex rational expression by writing it as division: $\frac{n - \frac{4 n}{n + 5}}{\frac{1}{n + 5} + \frac{1}{n - 5}} .$

Solution

	$A complex fraction with a numerator of n minus 4n over n plus 5, and a denominator of 1 over n plus 5 plus 1 over n minus 5.$
Simplify the numerator and denominator. Find common denominators for the numerator and denominator.	$An algebraic expression showing a complex fraction. The numerator involves subtraction of two fractions, and the denominator involves addition of two fractions. Variables 'n' are present.$
Simplify the numerators.	$A large algebraic fraction. The numerator is the difference of (n^2+5n)/(n+5) and 4n/(n+5). The denominator is the sum of (n-5)/((n+5)(n-5)) and (n+5)/((n-5)(n+5)).$
Subtract the rational expressions in the numerator and add in the denominator.	$A mathematical expression showing a complex algebraic fraction. The numerator is (n^2 + 5n - 4n) / (n + 5) and the denominator is (n - 5 + n + 5) / ((n + 5)(n - 5)).$
Simplify. (We now have one rational expression over one rational expression.)	$A complex algebraic fraction is displayed, with the numerator being (n^2 + n)/(n + 5) and the denominator being 2n/((n + 5)(n - 5)).$
Rewrite as fraction division.	$A mathematical expression showing the division of two algebraic fractions: (n^2 + n) / (n + 5) divided by (2n) / ((n + 5)(n - 5)).$
Multiply the first times the reciprocal of the second.	$A mathematical expression showing the product of two fractions: (n^2 + n) / (n + 5) multiplied by (n + 5)(n - 5) / (2n).$
Factor any expressions if possible.	$A mathematical expression showing a fraction with n(n+1)(n+5)(n-5) in the numerator and (n+5)2n in the denominator, set against a plain white background.$
Remove common factors.	$A fraction with algebraic terms in the numerator and denominator, showing cancellation of common factors 'n', '(n+5)', and '2' (implicitly from '2n').$
Simplify.	$A mathematical expression showing the fraction (n+1)(n-5) all divided by 2.$

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 the 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 the complex rational expression by using the LCD: $\frac{\frac{1}{3} + \frac{1}{6}}{\frac{1}{2} - \frac{1}{3}} .$

Solution

	$A complex fraction with (1/3 + 1/6) in the numerator and (1/2 - 1/3) in the denominator, illustrating basic fraction arithmetic.$
The LCD of all the fractions in the whole expression is 6.
Clear the fractions by multiplying the numerator and denominator by that LCD.	$A mathematical expression presented as a fraction. The numerator is 6 multiplied by the sum of 1/3 and 1/6. The denominator is 6 multiplied by the difference of 1/2 and 1/3. The '6's are red.$
Distribute.	$A mathematical expression showing a fraction where the numerator is (6 * 1/3) + (6 * 1/6) and the denominator is (6 * 1/2) - (6 * 1/3). The number 6 is highlighted in red in each term.$
Simplify.	$A mathematical expression showing a fraction with 2+1 in the numerator and 3-2 in the denominator.$
	$The fraction 3 over 1 is displayed in black text on a white background.$

We will use the same example as in Example 3. Decide which method works better for you.

How to Simplify a Complex Rational Expressing using the LCD

Simplify the complex rational expression by using the LCD: $\frac{\frac{1}{x} + \frac{1}{y}}{\frac{x}{y} - \frac{y}{x}} .$

Solution

Step 1 is to find the least common denominator of the complex rational expression, the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by the difference between the quantity x divided by y and the quantity y divided by x.

Step 2 is to multiply the numerator and the denominator by the least common denominator, x y. The result is x y times the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by x y times the difference between the quantity x divided by y and the quantity y divided by x.

Step 3 is to simplify the expression. Distribute x y in the numerator and the denominator. The result is x y times 1 divided by x plus x y times 1 divided by y all divided by x y times x divided by y plus x y times y divided by x. It simplifies to the sum of y and x divided by the quantity x squared minus y squared. Write the denominator as the difference of squares, the quantity x minus y times the quantity x plus y. The result is the quantity y plus x all divided by the quantity x minus y times the quantity x plus y. Remove the common factor, y plus x, from the numerator and denominator. The result is 1 divided by the quantity x minus y.

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

Simplify the complex rational expression by using the LCD: $\frac{\frac{2}{x + 6}}{\frac{4}{x - 6} - \frac{4}{x^{2} - 36}} .$

Solution

	$A complex fraction with 2/(x+6) in the numerator, and in the denominator, 4/(x-6) minus 4/(x^2-36).$
Find the LCD of all fractions in the complex rational expression. The LCD is $x^{2} - 36 = (x + 6) (x - 6)$ .
Multiply the numerator and denominator by the LCD.	$An algebraic fraction where both numerator and denominator are multiplied by (x+6)(x-6) as a step to simplify the expression and eliminate nested fractions.$
Simplify the expression.
Distribute in the denominator.	$A mathematical expression displaying a fraction. The numerator has (x+6)(x-6) times 2/(x+6). The denominator subtracts two terms, both starting with (x+6)(x-6).$
Simplify.	$A complex algebraic fraction showing several terms being canceled out, including (x+6) and (x-6), across the numerator and the two subtracted terms in the denominator.$
Simplify.	$A mathematical expression showing the fraction 2(x - 6) divided by 4(x + 6) - 4.$
To simplify the denominator, distribute and combine like terms.	$A mathematical expression showing a fraction with 2(x - 6) in the numerator and 4x + 20 in the denominator.$
Factor the denominator.	$A mathematical fraction with 2(x-6) in the numerator and 4(x+5) in the denominator.$
Remove common factors.	$A mathematical fraction showing the simplification of 2(x-6) over 2*2(x+5), with one '2' in the numerator and one in the denominator crossed out.$
Simplify.	$A mathematical expression displaying a fraction with (x - 6) in the numerator and 2(x + 5) in the denominator.$
Notice that there are no more factors common to the numerator and denominator.

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

Simplify the complex rational expression by using the LCD: $\frac{\frac{4}{m^{2} - 7 m + 12}}{\frac{3}{m - 3} - \frac{2}{m - 4}} .$

Solution

	$A complex algebraic fraction with 4 over the quadratic m^2 - 7m + 12 in the numerator, and the difference of two rational expressions, 3/(m-3) - 2/(m-4), in the denominator.$
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.	$Algebraic expression demonstrating the simplification of a complex fraction by multiplying both the numerator and denominator by (m-3)(m-4).$
Simplify.	$A complex algebraic fraction showing cancellation of terms (m-3) and (m-4) in both the numerator and denominator, simplifying to 4 over (m-4)3 - (m-3)2.$
Simplify.	$A mathematical expression showing the fraction 4 divided by the quantity 3 times (m minus 4) minus 2 times (m minus 3).$
Distribute.	$A mathematical expression showing the fraction 4 over the quantity 3m minus 12 minus 2m plus 6, which simplifies to 4 over m minus 6.$
Combine like terms.	$A fraction is displayed with a numerator of 4 and a denominator of m minus 6.$

Simplify the complex rational expression by using the LCD: $\frac{\frac{y}{y + 1}}{1 + \frac{1}{y - 1}} .$

Solution

	$A complex fraction with y/(y+1) in the numerator and 1 + 1/(y-1) in the denominator, representing a rational algebraic expression.$
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.	$An algebraic complex fraction. The numerator is (y+1)(y-1)y/(y+1) and the denominator is (y+1)(y-1)(1 + 1/(y-1)). The factor (y+1)(y-1) is shown in red.$
Distribute in the denominator and simplify.	$Simplifying a rational algebraic expression by cancelling common factors (y+1) and (y-1) from the numerator and denominator of the fraction.$
Simplify.	$A mathematical fraction with (y-1)y in the numerator and (y+1)(y-1) + (y+1) in the denominator.$
Simplify the denominator and leave the numerator factored.	$A mathematical expression displaying a fraction. The numerator is y(y-1) and the denominator is y^2 - 1 + y + 1.$
	$A mathematical expression showing the fraction y(y-1) over y^2 + y.$
Factor the denominator and remove factors common with the numerator.	$A mathematical fraction is shown, with y(y-1) as the numerator and y(y+1) as the denominator.$
Simplify.	$A mathematical expression showing the fraction (y - 1) / (y + 1). The numerator is 'y minus 1' and the denominator is 'y plus 1', separated by a horizontal fraction bar.$

Key Concepts

How to 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.
How 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 each complex rational expression by writing it as division.

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

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 each complex rational expression by using the LCD.

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

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

$\frac{m (m - 5)}{(4 m - 19) (m + 5)}$

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

In the following exercises, simplify each complex rational expression using 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

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

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

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 or 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, $\frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a} - \frac{1}{b}} .$ 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.

This table has four columns and three rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was simplify a complex rational expression by writing it as division. In row 3, the I can was simplify a complex rational expression by using the least common denominator.

ⓑ 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 and/or denominator contains a rational expression.