8.9: Complex Rational Expressions

Simple And Complex Fractions

Simple Fraction

In section 8.2 we saw that a simple fraction was a fraction of the form $\dfrac{P}{Q}$, where $P$ and $Q$ are polynomials and $Q \not = 0$.

Complex Fraction

A complex fraction is a fraction in which the numerator or denominator, or both, is a fraction. The fractions

$\dfrac{\frac{8}{15}}{\frac{2}{3}}$ and $\dfrac{1 - \frac{1}{x}}{1 - \frac{1}{x^2}}$

are examples of complex fractions, or more generally, complex rational expressions.

There are two methods for simplifying complex rational expressions: the combine-divide method and the LCD-multiply-divide method.

The Combine-Divide Method

Combine-Divide Method

1. If necessary, combine the terms of the numerator together.
2. If necessary, combine the terms of the denominator together.
3. Divide the numerator by the denominator.

Sample Set A

Simplify each complex rational expression.

Example $\PageIndex{1}$

$\dfrac{\frac{x^3}{8}}{\frac{x^5}{12}}$

Steps 1 and 2 are not necessary so we proceed with step 3:

$\dfrac{\frac{x^3}{8}}{\frac{x^5}{12}} = \dfrac{x^3}{8} \cdot \dfrac{12}{x^5} = \dfrac{\cancel{x^3}}{^\cancel{8}_2} \cdot \dfrac{_\cancel{12}^3}{x^{\cancel{5}2}} = \dfrac{3}{2x^2}$

Example $\PageIndex{2}$

$\dfrac{1 - \frac{1}{x}}{1 - \frac{1}{x^2}}$

Step 1: Combine the terms of the numerator: LCD = $x$.

$1 - \dfrac{1}{x} = \dfrac{x}{x} - \dfrac{1}{x} = \dfrac{x-1}{x}$

Step 2: Combine the terms of the denominator: LCD = $x^2$.

$1 - \dfrac{1}{x^2} = \dfrac{x^2}{x^2} - \dfrac{1}{x^2} = \dfrac{x^2 - 1}{x^2}$

Step 3: Divide the numerator by the denominator.

$\begin{array}{flushleft} \dfrac{\frac{x-1}{x}}{\frac{x^2-1}{x^2}} &= \dfrac{x-1}{x} \cdot \dfrac{x^2}{x^2-1}\\ &= \dfrac{\cancel{x-1}}{\cancel{x}} \dfrac{x^{\cancel{2}}}{(x+1)\cancel{(x+1)}}\\ &= \dfrac{x}{x+1} \end{array}$

Thus,

$\dfrac{1 - \frac{1}{x}}{1 - \frac{1}{x^2}} = \dfrac{x}{x+1}$

Example $\PageIndex{3}$

$\dfrac{2 - \frac{13}{m} - \frac{7}{m^2}}{2 + \frac{3}{m} + \frac{1}{m^2}}$

Step 1: Combine the terms of the numerator: LCD = $m^2$.

$2-\dfrac{13}{m}-\dfrac{7}{m^{2}}=\dfrac{2 m^{2}}{m^{2}}-\dfrac{13 m}{m^{2}}-\dfrac{7}{m^{2}}=\dfrac{2 m^{2}-13 m-7}{m^{2}}$

Step 2: Combine the terms of the denominator: LCD = $m^2$

$2+\dfrac{3}{m}+\dfrac{1}{m^{2}}=\dfrac{2 m^{2}}{m^{2}}+\dfrac{3 m}{m^{2}}+\dfrac{1}{m^{2}}=\dfrac{2 m^{2}+3 m+1}{m^{2}}$

Step 3: Divide the numerator by the denominator:

$\begin{array}{flushleft} \dfrac{\frac{2 m^{2}-13 m-7}{m^{2}}}{\frac{2 m^{2}+3 m-1}{m^{2}}} &=\dfrac{2 m^{2}-13 m-7}{m^{2}} \cdot \frac{m^{2}}{2 m^{2}+3 m+1} \\ &=\dfrac{\cancel{(2 m+1)}(m-7)}{\cancel{m^2}} \cdot \dfrac{\cancel{m^2}}{\cancel{(2 m+1)}(m+1)} \\ &=\dfrac{m-7}{m+1} \end{array}$

Thus,

$\dfrac{2 - \frac{13}{m} - \frac{7}{m^2}}{2 + \frac{3}{m} + \frac{1}{m^2}} = \dfrac{m - 7}{m + 1}$

Practice Set A

Use the combine-divide method to simplify each expression.

Practice Problem $\PageIndex{1}$

$\dfrac{\frac{27x^2}{6}}{\frac{15x^3}{8}}$

Answer

$\dfrac{12}{5x}$

Practice Problem $\PageIndex{2}$

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

Answer

$\dfrac{3x - 1}{3x + 1}$

Practice Problem $\PageIndex{3}$

$\dfrac{1 + \frac{x}{y}}{x - \frac{y^2}{x}}$

Answer

$\dfrac{x}{y(x-y)}$

Practice Problem $\PageIndex{4}$

$\dfrac{m - 3 + \frac{2}{m}}{m - 4 + \frac{3}{m}}$

Answer

$\dfrac{m-2}{m-3}$

Practice Problem $\PageIndex{5}$

$\dfrac{1 + \frac{1}{x-1}}{1 - \frac{1}{x-1}}$

Answer

$\dfrac{x}{x-2}$

The LCD-Multiply-Divide Method

LCD-Multiply-Divide Method

1. Find the LCD of all the terms.
2. Multiply the numerator and denominator by the LCD.
3. Reduce if necessary.

Sample Set B

Simplify each complex fraction.

Example $\PageIndex{4}$

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

Step 1: The LCD $=a^2$.

Step 2: Multiply both the numerator and denominator by $a^2$.

$\begin{array}{flushleft} \dfrac{a^2(1 - \frac{4}{a^2})}{a^2(1 + \frac{2}{a})} &= \dfrac{a^2 \cdot 1-a^2 \cdot \frac{4}{a^2}}{a^2 \cdot 1+a^2\cdot\frac{2}{a}}\\ &= \dfrac{a^2-4}{a^2 + 2a} \end{array}$.

Step 3: Reduce:

$\begin{array}{flushleft} \frac{a^{2}-4}{a^{2}+2 a} &=\frac{\cancel{(a+2)}(a-2)}{a\cancel{(a+2)}} \\ &=\frac{a-2}{a} \end{array}$

Thus,

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

Example $\PageIndex{5}$

$\dfrac{1 - \frac{5}{x} - \frac{6}{x^2}}{1 + \frac{6}{x} + \frac{5}{x^2}}$

Step 1: The LCD is $x^2$.

Step 2: Multiply the numerator and denominator by $x^2$.

$\begin{array}{flushleft} \dfrac{x^{2}(1-\frac{5}{x}-\frac{6}{x^{2}})}{x^{2}(1+\frac{6}{x}+\frac{5}{x^{2}})} &= \dfrac{x^{2} \cdot 1-x^{\cancel{2}} \cdot \frac{5}{\cancel{x}}-\cancel{x^{2}} \cdot \frac{6}{\cancel{x^{2}}}}{x^{2} \cdot 1+x^{\cancel{2}} \cdot \frac{6}{\cancel{x}}+\cancel{x^2} \cdot \frac{5}{\cancel{x^2}}} \\ &=\dfrac{x^{2}-5 x-6}{x^{2}+6 x+5} \end{array}$

Step 3: Reduce:

$\begin{array}{flushleft} \dfrac{x^{2}-5 x-6}{x^{2}+6 x+5} &=\dfrac{(x-6)(x+1)}{(x+5)(x+1)} \\ &=\dfrac{x-6}{x+5} \end{array}$

Thus,

$\dfrac{1 - \frac{5}{x} - \frac{6}{x^2}}{1 + \frac{6}{x} + \frac{5}{x^2}} = \dfrac{x-6}{x+5}$

Practice Set B

The following problems are the same problems as the problems in Practice Set A. Simplify these expressions using the LCD-multiply-divide method. Compare the answers to the answers produced in Practice Set A.

Practice Problem $\PageIndex{6}$

$\dfrac{\frac{27x^2}{6}}{\frac{15x^3}{8}}$

Answer

$\dfrac{12}{5x}$

Practice Problem $\PageIndex{7}$

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

Answer

$\dfrac{3x - 1}{3x + 1}$

Practice Problem $\PageIndex{8}$

$\dfrac{1 + \frac{x}{y}}{x - \frac{y^2}{x}}$

Answer

$\dfrac{x}{y(x-y)}$

Practice Problem $\PageIndex{9}$

$\dfrac{m - 3 + \frac{2}{m}}{m - 4 + \frac{3}{m}}$

Answer

$\dfrac{m-2}{m-3}$

Practice Problem $\PageIndex{10}$

$\dfrac{1 + \frac{1}{x-1}}{1 - \frac{1}{x-1}}$

Answer

$\dfrac{x}{x-2}$

Exercises

For the following problems, simplify each complex rational expression.

Exercise $\PageIndex{1}$

$\dfrac{1+\frac{1}{4}}{1-\frac{1}{4}}$

Answer

$\dfrac{5}{3}$

Exercise $\PageIndex{2}$

$\dfrac{1-\frac{1}{3}}{1+\frac{1}{3}}$

Exercise $\PageIndex{3}$

$\dfrac{1-\frac{1}{y}}{1+\frac{1}{y}}$

Answer

$\dfrac{y-1}{y+1}$

Exercise $\PageIndex{4}$

$\dfrac{a+\frac{1}{x}}{a-\frac{1}{x}}$

Exercise $\PageIndex{5}$

$\dfrac{\frac{a}{b}+\frac{c}{b}}{\frac{a}{b}-\frac{c}{b}}$

Answer

$\dfrac{a+c}{a-c}$

Exercise $\PageIndex{6}$

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

Exercise $\PageIndex{7}$

$\dfrac{3+\frac{1}{x}}{\frac{3 x+1}{x^{2}}}$

Answer

$x$

Exercise $\PageIndex{8}$

$\dfrac{1+\frac{x}{x+y}}{1-\frac{x}{x+y}}$

Exercise $\PageIndex{9}$

$\dfrac{2+\frac{5}{a+1}}{2-\frac{5}{a+1}}$

Answer

$\dfrac{2a + 7}{2a - 3}$

Exercise $\PageIndex{10}$

$\dfrac{1-\frac{1}{a-1}}{1+\frac{1}{a-1}}$

Exercise $\PageIndex{11}$

$\dfrac{4-\frac{1}{m^{2}}}{2+\frac{1}{m}}$

Answer

$\dfrac{2m - 1}{m}$

Exercise $\PageIndex{12}$

$\dfrac{9-\frac{1}{x^{2}}}{3-\frac{1}{x}}$

Exercise $\PageIndex{13}$

$\dfrac{k-\frac{1}{k}}{\frac{k+1}{k}}$

Answer

$k-1$

Exercise $\PageIndex{14}$

$\dfrac{\frac{m}{m+1}-1}{\frac{m+1}{2}}$

Exercise $\PageIndex{15}$

$\dfrac{\frac{2 x y}{2 x-y}-y}{\frac{2 x-y}{3}}$

Answer

$\dfrac{3y^2}{(2x - y)^2}$

Exercise $\PageIndex{16}$

$\dfrac{\frac{1}{a+b}-\frac{1}{a-b}}{\frac{1}{a+b}+\frac{1}{a-b}}$

Exercise $\PageIndex{17}$

$\dfrac{\frac{5}{x+3}-\frac{5}{x-3}}{\frac{5}{x+3}+\frac{5}{x-3}}$

Answer

$\dfrac{-3}{x}$

Exercise $\PageIndex{18}$

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

Exercise $\PageIndex{19}$

$\dfrac{\frac{1}{x^{2}}-\frac{1}{y^{2}}}{\frac{1}{x}+\frac{1}{y}}$

Answer

$\dfrac{y-x}{xy}$

Exercise $\PageIndex{20}$

$\dfrac{1+\frac{5}{x}+\frac{6}{x^{2}}}{1-\frac{1}{x}-\frac{12}{x^{2}}}$

Exercise $\PageIndex{21}$

$\dfrac{1+\frac{1}{y}-\frac{2}{y^{2}}}{1+\frac{7}{y}+\frac{10}{y^{2}}}$

Answer

$\dfrac{y-1}{y+5}$

Exercise $\PageIndex{22}$

$\dfrac{\frac{3 n}{m}-2-\frac{m}{n}}{\frac{3 n}{m}+4+\frac{m}{n}}$

Answer

$3x−4$

Exercise $\PageIndex{23}$

$\dfrac{\frac{y}{x+y}-\frac{x}{x-y}}{\frac{x}{x+y}+\frac{y}{x-y}}$

Exercise $\PageIndex{24}$

$\dfrac{\frac{a}{a-2}-\frac{a}{a+2}}{\frac{2 a}{a-2}+\frac{a^{2}}{a+2}}$

Answer

$\dfrac{4}{a^2 + 4}$

Exercise $\PageIndex{25}$

$3 - \dfrac{2}{1 - \frac{1}{m+1}}$

Exercise $\PageIndex{26}$

$\dfrac{x-\frac{1}{1-\frac{1}{x}}}{x+\frac{1}{1+\frac{1}{x}}}$

Answer

$\dfrac{(x-2)(x+1)}{(x-1)(x+2)}$

Exercise $\PageIndex{27}$

In electricity theory, when two resistors of resistance $R_1$ and $R_2$ ohms are connected in parallel, the total resistance $R$ is:

$R = \dfrac{1}{\frac{1}{R_1} + \frac{1}{R_2}}$

Write this complex fraction as a simple fraction.

Exercise $\PageIndex{28}$

According to Einstein's theory of relativity, two velocities $v_1$ and $v_2$ are not added according to $v = v_1 + v_2$, but rather by

$v = \dfrac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}}$

Write this complex fraction as a simple fraction.

Einstein's formula is really only applicable for velocities near the speed of light ($c=186,000$ miles per second). At very much lower velocities, such as 500 miles per hour, the formula $v=v_1+v_2$ provides an extremely good approximation.

Answer

$\dfrac{c^2(V_1 + V_2)}{c^2 + V_1V_2}$

Exercises For Review

Exercise $\PageIndex{30}$

Supply the missing word. Absolute value speaks to the question of how ____ and not “which way.”

Exercise $\PageIndex{31}$

Find the product. $(3x + 4)^2$

Answer

$9x^2 + 24x + 16$

Exercise $\PageIndex{32}$

Factor $x^4 - y^4$

Exercise $\PageIndex{33}$

Solve the equation $\dfrac{3}{x-1} - \dfrac{5}{x+3} = 0$.

Answer

$x=7$

Exercise $\PageIndex{34}$

One inlet pipe can fill a tank in 10 minutes. Another inlet pipe can fill the same tank in 4 minutes. How long does it take both pipes working together to fill the tank?