8.8 Complex rational expressions

Elementary algebra Page 1 / 1

<para>This module is from<link document="col10614">Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr.</para><para>A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.</para><para>The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.</para><para>Objectives of this module: be able to distinguish between simple and complex fractions, be able to simplify complex fractions using the combine-divide and the LCD-multiply-divide method.</para>

Overview

Simple And Complex Fractions
The Combine-Divide Method
The LCD-Multiply-Divide Method

Simple and complex fractions

Simple fraction

In Section [link] we saw that a simple fraction was a fraction of the form

\frac{P}{Q},

where

P

and

Q

are polynomials and

Q \neq 0

Complex fraction

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

\begin{array}{l} \frac{\frac{8}{15}}{\frac{2}{3}} & and & \frac{1 - \frac{1}{x}}{1 - \frac{1}{x^{2}}} \end{array}

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

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

Sample set a

Simplify each complex rational expression.

$\begin{array}{l} \frac{\frac{x^{3}}{8}}{\frac{x^{5}}{12}} \\ Steps 1 and 2 are not necessary so we proceed with step 3 . \\ \frac{\frac{x^{3}}{8}}{\frac{x^{5}}{12}} = \frac{x^{3}}{8} \cdot \frac{12}{x^{5}} = \frac{x^{3}}{\underset{2}{8}} \cdot \frac{\overset{3}{12}}{x^{52}} = \frac{3}{2 x^{2}} \end{array}$

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$\begin{array}{l} \frac{1 - \frac{1}{x}}{1 - \frac{1}{x^{2}}} \\ Step 1: Combine the terms of the numerator: LCD = x . \\ 1 - \frac{1}{x} = \frac{x}{x} - \frac{1}{x} = \frac{x - 1}{x} \\ Step 2: Combine the terms of the denominator: LCD = x^{2} \\ 1 - \frac{1}{x^{2}} = \frac{x^{2}}{x^{2}} - \frac{1}{x^{2}} = \frac{x^{2} - 1}{x^{2}} \\ Step 3: Divide the numerator by the denominator . \\ \begin{array}{l} \frac{\frac{x - 1}{x}}{\frac{x^{2} - 1}{x^{2}}} & = & \frac{x - 1}{x} \cdot \frac{x^{2}}{x^{2} - 1} \\ = & \frac{x - 1}{x} \frac{x^{2}}{(x + 1) (x - 1)} \\ = & \frac{x}{x + 1} \end{array} \\ Thus, \\ \frac{1 - \frac{1}{x}}{1 - \frac{1}{x^{2}}} = \frac{x}{x + 1} \end{array}$

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$\begin{array}{l} \frac{2 - \frac{13}{m} - \frac{7}{m^{2}}}{2 + \frac{3}{m} + \frac{1}{m^{2}}} \\ Step 1: Combine the terms of the numerator: LCD = m^{2} . \\ 2 - \frac{13}{m} - \frac{7}{m^{2}} = \frac{2 m^{2}}{m^{2}} - \frac{13 m}{m^{2}} - \frac{7}{m^{2}} = \frac{2 m^{2} - 13 m - 7}{m^{2}} \\ Step 2: Combine the terms of the denominator: LCD = m^{2} \\ 2 + \frac{3}{m} + \frac{1}{m^{2}} = \frac{2 m^{2}}{m^{2}} + \frac{3 m}{m^{2}} + \frac{1}{m^{2}} = \frac{2 m^{2} + 3 m + 1}{m^{2}} \\ Step 3: Divide the numerator by the denominator . \\ \begin{array}{l} \frac{\frac{2 m^{2} - 13 m - 7}{m^{2}}}{\frac{2 m^{2} + 3 m - 1}{m^{2}}} & = & \frac{2 m^{2} - 13 m - 7}{m^{2}} \cdot \frac{m^{2}}{2 m^{2} + 3 m + 1} \\ = & \frac{(2 m + 1) (m - 7)}{m^{2}} \cdot \frac{m^{2}}{(2 m + 1) (m + 1)} \\ = & \frac{m - 7}{m + 1} \end{array} \\ Thus, \\ \frac{2 - \frac{13}{m} - \frac{7}{m^{2}}}{2 + \frac{3}{m} + \frac{1}{m^{2}}} = \frac{m - 7}{m + 1} \end{array}$

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Practice set a

Use the combine-divide method to simplify each expression.

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

$\frac{12}{5 x}$

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

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

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

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

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

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

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

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

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The lcd-multiply-divide method

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

Sample set b

Simplify each complex fraction.

$\begin{array}{l} \frac{1 - \frac{4}{a^{2}}}{1 + \frac{2}{a}} \\ Step 1: The LCD = a^{2} . \\ Step 2: Multiply both the numerator and denominator by a^{2} . \\ \begin{array}{l} \frac{a^{2} (1 - \frac{4}{a^{2}})}{a^{2} (1 + \frac{2}{a})} & = & \frac{a^{2} \cdot 1 - a^{2} \cdot \frac{4}{a^{2}}}{a^{2} \cdot 1 + a^{2} \cdot \frac{2}{a}} \\ = & \frac{a^{2} - 4}{a^{2} + 2 a} \end{array} \\ Step 3: Reduce . \\ \begin{array}{l} \frac{a^{2} - 4}{a^{2} + 2 a} & = & \frac{(a + 2) (a - 2)}{a (a + 2)} \\ = & \frac{a - 2}{a} \end{array} \\ Thus, \\ \frac{1 - \frac{4}{a^{2}}}{1 + \frac{2}{a}} = \frac{a - 2}{a} \end{array}$

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$\begin{array}{l} \begin{array}{l} \frac{1 - \frac{5}{x} - \frac{6}{x^{2}}}{1 + \frac{6}{x} + \frac{5}{x^{2}}} \\ Step 1 : The LCD is x^{2} . \\ Step 2 : Multiply the numerator and denominator by x^{2} . \\ \begin{array}{l} \frac{x^{2} (1 - \frac{5}{x} - \frac{6}{x^{2}})}{x^{2} (1 + \frac{6}{x} + \frac{5}{x^{2}})} & = & \frac{x^{2} \cdot 1 - x^{2} \cdot \frac{5}{x} - x^{2} \cdot \frac{6}{x^{2}}}{x^{2} \cdot 1 + x^{2} \cdot \frac{6}{x} + x^{2} \cdot \frac{5}{x^{2}}} \\ = & \frac{x^{2} - 5 x - 6}{x^{2} + 6 x + 5} \end{array} \end{array} \\ Step 3 : Reduce . \\ \begin{array}{l} \frac{x^{2} - 5 x - 6}{x^{2} + 6 x + 5} & = & \frac{(x - 6) (x + 1)}{(x + 5) (x + 1)} \\ = & \frac{x - 6}{x + 5} \end{array} \\ Thus, \\ \frac{1 - \frac{5}{x} - \frac{6}{x^{2}}}{1 + \frac{6}{x} + \frac{5}{x^{2}}} = \frac{x - 6}{x + 5} \end{array}$

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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.

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

$\frac{12}{5 x}$

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

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

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

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

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

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

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

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

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Exercises

For the following problems, simplify each complex rational expression.

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

$\frac{5}{3}$

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

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

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

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

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

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

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

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

$x$

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

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

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

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

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

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

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

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

$k - 1$

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

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

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

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

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

$\frac{- 3}{x}$

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

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

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

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

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

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

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

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$\frac{x - \frac{4}{3 x - 1}}{1 - \frac{2 x - 2}{3 x - 1}}$

$3 x - 4$

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

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

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

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

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

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

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In electricity theory, when two resistors of resistance $R_{1}$ and $R_{2}$ ohms are connected in parallel, the total resistance $R$ is

$R = \frac{1}{\frac{1}{R_{1}} + \frac{1}{R_{2}}}$

Write this complex fraction as a simple fraction.

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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

$v = \frac{v_{1} + v_{2}}{1 + \frac{v_{1} v_{2}}{c^{2}}}$

Write this complex fraction as a simple fraction.

Einstein's formula is really only applicale 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.

$\frac{c^{2} (V_{1} + V_{2})}{c^{2} + V_{1} V_{2}}$

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Exercises for review

( [link] ) Supply the missing word. Absolute value speaks to the question of how and not “which way.”

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( [link] ) Find the product. ${(3 x + 4)}^{2} .$

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

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( [link] ) Factor $x^{4} - y^{4} .$

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( [link] ) Solve the equation $\frac{3}{x - 1} - \frac{5}{x + 3} = 0.$

$x = 7$

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( [link] ) 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?

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Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

how can I help

Sir

hmm can we speak here?

Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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Source: OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3

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