# 8.9: Complex Rational Expressions

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## 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 $$\PageIndex{1}$$

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

$$\dfrac{12}{5x}$$

##### Practice Problem $$\PageIndex{2}$$

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

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

##### Practice Problem $$\PageIndex{3}$$

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

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

##### Practice Problem $$\PageIndex{4}$$

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

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

##### Practice Problem $$\PageIndex{5}$$

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

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

## 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 $$\PageIndex{6}$$

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

$$\dfrac{12}{5x}$$

##### Practice Problem $$\PageIndex{7}$$

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

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

##### Practice Problem $$\PageIndex{8}$$

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

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

##### Practice Problem $$\PageIndex{9}$$

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

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

##### Practice Problem $$\PageIndex{10}$$

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

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

## Exercises

For the following problems, simplify each complex rational expression.

##### Exercise $$\PageIndex{1}$$

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

$$\dfrac{5}{3}$$

##### Exercise $$\PageIndex{2}$$

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

##### Exercise $$\PageIndex{3}$$

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

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

##### Exercise $$\PageIndex{4}$$

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

##### Exercise $$\PageIndex{5}$$

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

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

##### Exercise $$\PageIndex{6}$$

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

##### Exercise $$\PageIndex{7}$$

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

$$x$$

##### Exercise $$\PageIndex{8}$$

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

##### Exercise $$\PageIndex{9}$$

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

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

##### Exercise $$\PageIndex{10}$$

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

##### Exercise $$\PageIndex{11}$$

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

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

##### Exercise $$\PageIndex{12}$$

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

##### Exercise $$\PageIndex{13}$$

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

$$k-1$$

##### Exercise $$\PageIndex{14}$$

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

##### Exercise $$\PageIndex{15}$$

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

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

##### Exercise $$\PageIndex{16}$$

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

##### Exercise $$\PageIndex{17}$$

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

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

##### Exercise $$\PageIndex{18}$$

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

##### Exercise $$\PageIndex{19}$$

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

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

##### Exercise $$\PageIndex{20}$$

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

##### Exercise $$\PageIndex{21}$$

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

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

##### Exercise $$\PageIndex{22}$$

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

$$3x−4$$

##### Exercise $$\PageIndex{23}$$

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

##### Exercise $$\PageIndex{24}$$

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

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

##### Exercise $$\PageIndex{25}$$

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

##### Exercise $$\PageIndex{26}$$

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

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

##### 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.

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

## 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. $$(3x + 4)^2$$

$$9x^2 + 24x + 16$$

##### Exercise $$\PageIndex{32}$$

Factor $$x^4 - y^4$$

##### Exercise $$\PageIndex{33}$$

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

$$x=7$$