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

differentiate between demand and supply giving examples

Lambiv Reply

differentiated between demand and supply using examples

Lambiv

what is labour ?

Lambiv

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Eliyee

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Lambiv

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Aster Reply

appreciation

Eliyee

explain perfect market

Lindiwe Reply

In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

Ezea

What is ceteris paribus?

Shukri Reply

other things being equal

AI-Robot

When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab

Kelo

Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)

Kelo

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Shukri

Can I ask you other question?

Shukri

what is monopoly mean?

Habtamu Reply

What is different between quantity demand and demand?

Shukri Reply

Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded

Ezea

Shukri

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Lilia Reply

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Fiker Reply

Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.

Shukri

production function means

Jabir

What do you think is more important to focus on when considering inequality ?

Abdisa Reply

any question about economics?

Awais Reply

sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏

Asui

it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

Awais

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Asui

In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p

Cornelius

Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .

Feyisa Reply

Answer

Feyisa

Jabir

the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema

Gsbwnw Reply

suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product

Abdureman

types of unemployment

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What is the difference between perfect competition and monopolistic competition?

Mohammed

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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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