Algebra trigonometry 8e ron larson

Set of natural numbers Set of whole numbers Set of integers A real number is rational if it can be written as the ratio of two integers, whereFor instance, the numbers andare rational..

Algebra and Trigonometry

Eighth Edition

Ron Larson

The Pennsylvania State University

The Behrend College

With the assistance of

David C Falvo

The Pennsylvania State University

The Behrend College

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A Word from the Author (Preface) vii

Prerequisites 1

Contents

chapter P

chapter 1

chapter 2

Polynomial Functions 259

chapter 3

chapter 4

chapter 5

chapter 6

Contents v

chapter 7

chapter 8

chapter 9

chapter 10

chapter 11

Index of Applications (web) Appendix B Concepts in Statistics (web)

In the Eighth Edition, we continue to offer instructors and students a text that is pedagogically sound, mathematically precise, and still comprehensible There are manychanges in the mathematics, art, and design; the more significant changes are noted here

• New Chapter Openers Each Chapter Opener has three parts, In Mathematics, In

Real Life, and In Careers In Mathematics describes an important mathematical

topic taught in the chapter In Real Life tells students where they will encounter this topic in real-life situations In Careers relates application exercises to a variety of

careers

students in two new features The Study Tip provides students with useful information or suggestions for learning the topic The Warning/Caution points out

common mathematical errors made by students

• New Algebra Helps Algebra Help directs students to sections of the textbook

where they can review algebra skills needed to master the current topic

• New Side-by-Side Examples Throughout the text, we present solutions to manyexamples from multiple perspectives—algebraically, graphically, and numerically.The side-by-side format of this pedagogical feature helps students to see that a problemcan be solved in more than one way and to see that different methods yield the sameresult The side-by-side format also addresses many different learning styles

A Word from

the Author

Welcome to the Eighth Edition of Algebra and Trigonometry! We are proud to offer you

a new and revised version of our textbook With this edition, we have listened to you,our users, and have incorporated many of your suggestions for improvement

• New Capstone Exercises Capstones are conceptual problems that synthesize key

topics and provide students with a better understanding of each section’s concepts Capstone exercises are excellent for classroom discussion or test prep, andteachers may find value in integrating these problems into their reviews of the section

and/or example of each objective taught in the chapter

• Revised Exercise Sets The exercise sets have been carefully and extensivelyexamined to ensure they are rigorous and cover all topics suggested by our users.Many new skill-building and challenging exercises have been added

For the past several years, we’ve maintained an independent website—

CalcChat.com—that provides free solutions to all odd-numbered exercises in the text.

Thousands of students using our textbooks have visited the site for practice and helpwith their homework For the Eighth Edition, we were able to use information fromCalcChat.com, including which solutions students accessed most often, to help guidethe revision of the exercises

I hope you enjoy the Eighth Edition of Algebra and Trigonometry As always, I

welcome comments and suggestions for continued improvements

I would like to thank the many people who have helped me prepare the text and the supplements package Their encouragement, criticisms, and suggestions have beeninvaluable

Thank you to all of the instructors who took the time to review the changes in thisedition and to provide suggestions for improving it Without your help, this book wouldnot be possible

Reviewers

Chad Pierson, University of Minnesota-Duluth; Sally Shao, Cleveland State University;

Ed Stumpf, Central Carolina Community College; Fuzhen Zhang, Nova Southeastern

University; Dennis Shepherd, University of Colorado, Denver; Rhonda Kilgo, Jacksonville State University; C Altay Özgener, Manatee Community College Bradenton; William Forrest, Baton Rouge Community College; Tracy Cook, University

of Tennessee Knoxville; Charles Hale, California State Poly University Pomona; Samuel

Evers, University of Alabama; Seongchun Kwon, University of Toledo; Dr Arun K Agarwal, Grambling State University; Hyounkyun Oh, Savannah State University; Michael J McConnell, Clarion University; Martha Chalhoub, Collin County

Community College; Angela Lee Everett, Chattanooga State Tech Community College;

Heather Van Dyke, Walla Walla Community College; Gregory Buthusiem, Burlington

County Community College; Ward Shaffer, College of Coastal Georgia; Carmen

Thomas, Chatham University; Emily J Keaton

My thanks to David Falvo, The Behrend College, The Pennsylvania StateUniversity, for his contributions to this project My thanks also to Robert Hostetler, TheBehrend College, The Pennsylvania State University, and Bruce Edwards, University ofFlorida, for their significant contributions to previous editions of this text

I would also like to thank the staff at Larson Texts, Inc who assisted with reading the manuscript, preparing and proofreading the art package, and checking andtypesetting the supplements

proof-On a personal level, I am grateful to my spouse, Deanna Gilbert Larson, for herlove, patience, and support Also, a special thanks goes to R Scott O’Neil If you have suggestions for improving this text, please feel free to write to me Over the past twodecades I have received many useful comments from both instructors and students, and

I value these comments very highly

Ron Larson

Acknowledgments

Supplements for the Instructor

Annotated Instructor’s Edition This AIE is the complete student text plus use annotations for the instructor, including extra projects, classroom activities, teachingstrategies, and additional examples Answers to even-numbered text exercises,Vocabulary Checks, and Explorations are also provided

point-of-Complete Solutions Manual This manual contains solutions to all exercises from thetext, including Chapter Review Exercises and Chapter Tests

Instructor’s Companion Website This free companion website contains an abundance

of instructor resources

PowerLecture™ with ExamView® The CD-ROM provides the instructor with dynamicmedia tools for teaching college algebra PowerPoint® lecture slides and art slides ofthe figures from the text, together with electronic files for the test bank and a link to theSolution Builder, are available The algorithmic ExamView allows you to create, deliver,and customize tests (both print and online) in minutes with this easy-to-use assessmentsystem Enhance how your students interact with you, your lecture, and each other

Solutions Builder This is an electronic version of the complete solutions manualavailable via the PowerLecture and Instructor’s Companion Website It provides instructors with an efficient method for creating solution sets to homework or examsthat can then be printed or posted

Supplements xiSupplements for the Student

Student Companion Website This free companion website contains an abundance ofstudent resources

Instructional DVDs Keyed to the text by section, these DVDs provide comprehensivecoverage of the course—along with additional explanations of concepts, sample problems, and applications—to help students review essential topics

Student Study and Solutions Manual This guide offers step-by-step solutions for allodd-numbered text exercises, Chapter and Cumulative Tests, and Practice Tests withsolutions

Premium eBook The Premium eBook offers an interactive version of the textbookwith search features, highlighting and note-making tools, and direct links to videos ortutorials that elaborate on the text discussions

Enhanced WebAssign Enhanced WebAssign is designed for you to do your homework online This proven and reliable system uses pedagogy and content found inLarson’s text, and then enhances it to help you learn Algebra and Trigonometry moreeffectively Automatically graded homework allows you to focus on your learning andget interactive study assistance outside of class

Prerequisites

In Mathematics

Real numbers, exponents, radicals, and

polynomials are used in many different

branches of mathematics.

In Real Life

The concepts in this chapter are used to

model compound interest, volumes, rates

of change, and other real-life applications.

For instance, polynomials can be used

to model the stopping distance of an

automobile (See Exercise 116, page 36.)

IN CAREERS

There are many careers that use prealgebra concepts Several are listed below

• MeteorologistExercise 114, page 70

1

Real Numbers

Real numbers are used in everyday life to describe quantities such as age, miles per

gallon, and population Real numbers are represented by symbols such as

and

Here are some important subsets (each member of subset B is also a member of set A)

of the real numbers The three dots, called ellipsis points, indicate that the pattern

continues indefinitely

Set of natural numbers Set of whole numbers Set of integers

A real number is rational if it can be written as the ratio of two integers, whereFor instance, the numbers

andare rational The decimal representation of a rational number either repeats as in

or terminates as in A real number that cannot be written as the

ratio of two integers is called irrational Irrational numbers have infinite nonrepeating

decimal representations For instance, the numbers

andare irrational (The symbol means “is approximately equal to.”) Figure P.1 showssubsets of real numbers and their relationships to each other

Classifying Real Numbers

Determine which numbers in the set

are (a) natural numbers, (b) whole numbers, (c) integers, (d) rational numbers, and (e) irrational numbers

What you should learn

• Represent and classify real numbers.

• Order real numbers and use

inequalities.

• Find the absolute values of real

numbers and find the distance

between two real numbers.

• Evaluate algebraic expressions.

• Use the basic rules and

properties of algebra.

Why you should learn it

Real numbers are used to represent

many real-life quantities For example,

in Exercises 83– 88 on page 13, you

will use real numbers to represent the

Integers Noninteger

fractions (positive and negative)

Negative

integers

Whole numbers

Section P.1 Review of Real Numbers and Their Properties 3

Real numbers are represented graphically on the real number line When you draw a point on the real number line that corresponds to a real number, you are plotting the real number The point 0 on the real number line is the origin Numbers to the right

of 0 are positive, and numbers to the left of 0 are negative, as shown in Figure P.2 The

term nonnegative describes a number that is either positive or zero.

FIGURE P.2 The real number line

As illustrated in Figure P.3, there is a one-to-one correspondence between real

numbers and points on the real number line

Every real number corresponds to exactly Every point on the real number line

one point on the real number line corresponds to exactly one real number.

FIGURE P.3 One-to-one correspondence

Plotting Points on the Real Number Line

Plot the real numbers on the real number line

b The point representing the real number 2.3 lies between 2 and 3, but closer to 2, on

the real number line

c The point representing the real number lies between 0 and 1, butcloser to 1, on the real number line

d The point representing the real number lies between and but closer to

on the real number line Note that the point representing lies slightly tothe left of the point representing

Now try Exercise 17

Origin

−1 0 1 2

b a

FIGURE P.5 if and only if lies to

2

FIGURE P.9

Ordering Real Numbers

One important property of real numbers is that they are ordered.

Geometrically, this definition implies that if and only if lies to the left of

on the real number line, as shown in Figure P.5

Ordering Real Numbers

Place the appropriate inequality symbol or between the pair of real numbers

Solution

a Because lies to the left of 0 on the real number line, as shown in Figure P.6, youcan say that is less than 0, and write

b Because lies to the right of on the real number line, as shown in Figure P.7,

c Because lies to the left of on the real number line, as shown in Figure P.8, you

can say that is less than and write

d Because lies to the right of on the real number line, as shown in Figure P.9,

Now try Exercise 25

denotes all real numbers between and 3, including but not including 3, asshown in Figure P.11

Now try Exercise 31

1 2

1 5

1 3 1

1

4,

13

Definition of Order on the Real Number Line

If and are real numbers, is less than if is positive The order of and is denoted by the inequality This relationship can also be described

that is less than or equal to and the inequality means that is greater

than or equal to a.The symbols <,>,,and are inequality symbols.

b

b ≥ a b,

a

a ≤ b

b > a.

a b

b a

Inequalities can be used to describe subsets of real numbers called intervals In the bounded intervals below, the real numbers and are the endpoints of each interval.

The endpoints of a closed interval are included in the interval, whereas the endpoints of

an open interval are not included in the interval

real numbers They are simply convenient symbols used to describe the unboundedness

Using Inequalities to Represent Intervals

Use inequality notation to describe each of the following

Solution

a The statement “ is at most 2” can be represented by

b The statement “ is at least ” can be represented by

c “All in the interval ” can be represented by

Now try Exercise 45

Unbounded Intervals on the Real Number Line

Notation Interval Type Inequality Graph

x < b

, b

b x

Bounded Intervals on the Real Number Line

ClosedOpen

use a parenthesis and never a

bracket This is because and

are never an endpoint of an

interval and therefore are not

included in the interval

,



The reason that the four types

of intervals at the right are called

bounded is that each has a finite

length An interval that does not

have a finite length is unbounded

(see below)

Section P.1 Review of Real Numbers and Their Properties 5

Interpreting Intervals

Give a verbal description of each interval

Solution

a This interval consists of all real numbers that are greater than and less than 0

b This interval consists of all real numbers that are greater than or equal to 2.

c This interval consists of all negative real numbers.

Now try Exercise 41

Absolute Value and Distance

The absolute value of a real number is its magnitude, or the distance between the

origin and the point representing the real number on the real number line

Notice in this definition that the absolute value of a real number is never negative

number is either positive or zero Moreover, 0 is the only real number whose absolutevalue is 0 So,

Finding Absolute Values

Now try Exercise 51

Evaluating the Absolute Value of a Number

Example 6

Definition of Absolute Value

If is a real number, then the absolute value of is

a a, if a ≥ 0

a, if a < 0.

a a

The Law of Trichotomy states that for any two real numbers and precisely one

of three relationships is possible:

or Law of Trichotomy

Comparing Real Numbers

Place the appropriate symbol (<, >, or =) between the pair of real numbers

Solution

Now try Exercise 61

Absolute value can be used to define the distance between two points on the realnumber line For instance, the distance between and 4 is

as shown in Figure P.12

Finding a Distance

Solution

Distance between and 13

The distance can also be found as follows

Distance between and 13

Now try Exercise 67

Properties of Absolute Values

Distance Between Two Points on the Real Number Line

Let and be real numbers The distance between a and b is

b a

Algebraic Expressions

One characteristic of algebra is the use of letters to represent numbers The letters are

variables, and combinations of letters and numbers are algebraic expressions Here

are a few examples of algebraic expressions

The terms of an algebraic expression are those parts that are separated by addition.

For example,

numerical factor of a term is called the coefficient For instance, the coefficient of

is and the coefficient of is 1

Identifying Terms and Coefficients

Algebraic Expression Terms Coefficients

a.

b.

c.

Now try Exercise 89

To evaluate an algebraic expression, substitute numerical values for each of the

variables in the expression, as shown in the next example

Evaluating Algebraic Expressions

Note that you must substitute the value for each occurrence of the variable.

Now try Exercise 95

When an algebraic expression is evaluated, the Substitution Principle is used It

Example 12(a), for instance, 3 is substituted for in the expression x 3x 5.

a b

Definition of an Algebraic Expression

An algebraic expression is a collection of letters (variables) and real numbers (constants) combined using the operations of addition, subtraction, multiplication,

division, and exponentiation

Basic Rules of Algebra

There are four arithmetic operations with real numbers: addition, multiplication,

subtraction, and division, denoted by the symbols or and or / Of these, addition and multiplication are the two primary operations Subtraction anddivision are the inverse operations of addition and multiplication, respectively

Because the properties of real numbers below are true for variables and

algebraic expressions as well as for real numbers, they are often called the Basic Rules

of Algebra Try to formulate a verbal description of each property For instance, the

first property states that the order in which two real numbers are added does not affect

their sum.

,

, ,

Definitions of Subtraction and Division

Subtraction: Add the opposite Division: Multiply by the reciprocal.

If then

the multiplicative inverse (or reciprocal) of b In the fractional form

a is the numerator of the fraction and b is the denominator.

Basic Rules of Algebra

Let and be real numbers, variables, or algebraic expressions

Commutative Property of Addition:

Commutative Property of Multiplication:

Associative Property of Addition:

Associative Property of Multiplication:

Distributive Properties:

Additive Identity Property:

Multiplicative Identity Property:

Additive Inverse Property:

Because subtraction is defined as “adding the opposite,” the Distributive Properties

and division are neither commutative nor associative The examples

andshow that subtraction and division are not commutative Similarly

anddemonstrate that subtraction and division are not associative

Identifying Rules of Algebra

Identify the rule of algebra illustrated by the statement

a This statement illustrates the Commutative Property of Multiplication In other

words, you obtain the same result whether you multiply by 2, or 2 by

b This statement illustrates the Additive Inverse Property In terms of subtraction, this

property simply states that when any expression is subtracted from itself the result

is 0

c This statement illustrates the Multiplicative Inverse Property Note that it is

important that be a nonzero number If were 0, the reciprocal of would be undefined

d This statement illustrates the Associative Property of Addition In other words, to

form the sum

Now try Exercise 101

Properties of Negation and Equality

Let and be real numbers, variables, or algebraic expressions

Notice the difference between

the opposite of a number and a

negative number If is already

negative, then its opposite,

is positive For instance, if

Properties and Operations of Fractions

c Add fractions with unlike denominators:

Now try Exercise 119

A prime number is an integer that has exactly two positive factors—itself and 1—such

as 2, 3, 5, 7, and 11 The numbers 4, 6, 8, 9, and 10 are composite because each can be

written as the product of two or more prime numbers The number 1 is neither prime

nor composite The Fundamental Theorem of Arithmetic states that every

positive integer greater than 1 can be written as the product of prime numbers in

precisely one way (disregarding order) For instance, the prime factorization of 24 is

24 2223

c b

a

ab  c,

c b, a,

a 0,

a0 0

Properties and Operations of Fractions

Let a, b, c, and d be real numbers, variables, or algebraic expressions such that

and

1 Equivalent Fractions: if and only if

3 Generate Equivalent Fractions:

4 Add or Subtract with Like Denominators:

5 Add or Subtract with Unlike Denominators:

The “or” in the Zero-Factor

Property includes the possibility

that either or both factors may be

zero This is an inclusive or, and

it is the way the word “or” is

generally used in mathematics

In Property 1 of fractions, the

phrase “if and only if ” implies

two statements One statement

EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

P.1

VOCABULARY: Fill in the blanks

1 A real number is if it can be written as the ratio of two integers, where

2 numbers have infinite nonrepeating decimal representations.

3 The point 0 on the real number line is called the .

4 The distance between the origin and a point representing a real number on the real number line is

the of the real number

5 A number that can be written as the product of two or more prime numbers is called a number.

6 An integer that has exactly two positive factors, the integer itself and 1, is called a number.

7 An algebraic expression is a collection of letters called and real numbers called .

8 The of an algebraic expression are those parts separated by addition.

9 The numerical factor of a variable term is the of the variable term.

10 The states that if then or

SKILLS AND APPLICATIONS

In Exercises 11–16, determine which numbers in the set are

(a) natural numbers, (b) whole numbers, (c) integers,

(d) rational numbers, and (e) irrational numbers

In Exercises 19–22, use a calculator to find the decimal form

of the rational number If it is a nonterminating decimal,

write the repeating pattern

In Exercises 23 and 24, approximate the numbers and place

46. is at least and less than 0

47. is at least 10 and at most 22

48. is less than 5 but no less than

49 The dog’s weight is more than 65 pounds

50 The annual rate of inflation is expected to be at least

2.5% but no more than 5%

r W

3

k t

6

y

2

x y y

1, 2 2, 5

333

1 3 5

8

8 3

4.75

4 3

5.2

5 2 7

Section P.1 Review of Real Numbers and Their Properties 13

In Exercises 51– 60, evaluate the expression

between the two real numbers

73 The distance between and 5 is no more than 3.

74 The distance between and is at least 6

75. is at least six units from 0

76. is at most two units from

77 While traveling on the Pennsylvania Turnpike, you pass

milepost 57 near Pittsburgh, then milepost 236 near

Gettysburg How many miles do you travel during that

time period?

78 The temperature in Bismarck, North Dakota was

at noon, then at midnight What was the change in

temperature over the 12-hour period?

BUDGET VARIANCE In Exercises 79–82, the accounting

department of a sports drink bottling company is checking tosee whether the actual expenses of a department differ fromthe budgeted expenses by more than $500 or by more than5% Fill in the missing parts of the table, and determinewhether each actual expense passes the “budget variancetest.”

Budgeted Actual Expense, Expense,

FEDERAL DEFICIT In Exercises 83–88, use the bar graph,

which shows the receipts of the federal government (in billions of dollars) for selected years from 1996 through

2006 In each exercise you are given the expenditures of thefederal government Find the magnitude of the surplus ordeficit for the year (Source: U.S Office of Managementand Budget)

Year Receipts Expenditures Receipts

2025.5 1853.4

2407.3

1200 1400 1600 1800 2000 2200 2400 2600

1880.3 1722.0

n 1 0.5 0.01 0.0001 0.0000015

5

In Exercises 95–100, evaluate the expression for each value

of (If not possible, state the reason.)

In Exercises 113–120, perform the operation(s) (Write

fractional answers in simplest form.)

In Exercises 121 and 122, use the real numbers and

shown on the number line Determine the sign of each

expression

123 CONJECTURE

(a) Use a calculator to complete the table

(b) Use the result from part (a) to make a conjecture

124 CONJECTURE

(a) Use a calculator to complete the table

(b) Use the result from part (a) to make a conjecture

bound

TRUE OR FALSE? In Exercises 125–128, determine

whether the statement is true or false Justify your answer

130 THINK ABOUT IT Is there a difference between

saying that a real number is positive and saying that areal number is nonnegative? Explain

131 THINK ABOUT IT Because every even number is

divisible by 2, is it possible that there exist any evenprime numbers? Explain

132 THINK ABOUT IT Is it possible for a real number to

be both rational and irrational? Explain

real number a?Explain

a  a

v, u

Section P.2 Exponents and Radicals 15

Integer Exponents

Repeated multiplication can be written in exponential form.

Repeated Multiplication Exponential Form

What you should learn

• Use properties of exponents.

• Use scientific notation to represent

real numbers.

• Use properties of radicals.

• Simplify and combine radicals.

• Rationalize denominators and

numerators.

• Use properties of rational exponents.

Why you should learn it

Real numbers and algebraic expressions

are often written with exponents and

radicals For instance, in Exercise 121

on page 27, you will use an expression

involving rational exponents to find the

times required for a funnel to empty for

different water heights.

a n

a n

a n  aaa a

n a

Properties of Exponents

Let and be real numbers, variables, or algebraic expressions, and let and

be integers (All denominators and bases are nonzero.)

a

An exponent can also be negative In Property 3 below, be sure you see how to use anegative exponent

TECHNOLOGY

You can use a calculator to

evaluate exponential expressions

When doing so, it is important to

know when to use parentheses

because the calculator follows the

order of operations For instance,

The display will be 16 If you

omit the parentheses, the display

It is important to recognize the difference between expressions such as and

In the parentheses indicate that the exponent applies to the negative sign

and

The properties of exponents listed on the preceding page apply to all integers and

not just to positive integers, as shown in the examples in this section

Evaluating Exponential Expressions

a. Negative sign is part of the base.

b. Negative sign is not part of the base.

Now try Exercise 11

Evaluating Algebraic Expressions

Evaluate each algebraic expression when

Solution

a When the expression has a value of

Now try Exercise 23

Using Properties of Exponents

Use the properties of exponents to simplify each expression

24

24

Section P.2 Exponents and Radicals 17

Rewriting with Positive Exponents

Rewrite each expression with positive exponents

359,000,000,000,000,000,000

It is convenient to write such numbers in scientific notation This notation has the form

Earth can be written in scientific notation as

The positive exponent 20 indicates that the number is large (10 or more) and that the decimal point has been moved 20 places A negative exponent indicates that the number is small (less than 1) For instance, the mass (in grams) of one electron is

Rarely in algebra is there only

one way to solve a problem

Don’t be concerned if the steps

you use to solve a problem are

not exactly the same as the steps

presented in this text The

important thing is to use steps

that you understand and, of

course, steps that are justified

by the rules of algebra For

instance, you might prefer the

following steps for Example 4(d)

Note how Property 3 is used

in the first step of this solution

The fractional form of this

The French mathematician

Nicolas Chuquet (ca 1500)

wrote Triparty en la science des

nombres, in which a form of

exponent notation was used Our

expressions and were

and negative exponents were

also represented, so would be

10x2

6x3

To enter numbers in scientific notation, your calculator should have an exponential

entry key labeled

Begin by rewriting each number in scientific notation and simplifying

Now try Exercise 63(b)

2,400,000,000 0.0000045

Example 7

Radicals and Their Properties

A square root of a number is one of its two equal factors For example, 5 is a square root of 25 because 5 is one of the two equal factors of 25 In a similar way, a cube root

of a number is one of its three equal factors, as in

Some numbers have more than one nth root For example, both 5 and are square

roots of 25 The principal square root of 25, written as is the positive root, 5 The

principal nth root of a number is defined as follows.

A common misunderstanding is that the square root sign implies both negative andpositive roots This is not correct The square root sign implies only a positive root.When a negative root is needed, you must use the negative sign with the square rootsign

Evaluating Expressions Involving Radicals

Section P.2 Exponents and Radicals 19

Definition of nth Root of a Number

Let a and b be real numbers and let be a positive integer If

Principal nth Root of a Number

Let a be a real number that has at least one nth root The principal nth root of a

is the nth root that has the same sign as a It is denoted by a radical symbol

Principal nth root

The positive integer n is the index of the radical, and the number a is the radicand.

Here are some generalizations about the nth roots of real numbers.

Integers such as 1, 4, 9, 16, 25, and 36 are called perfect squares because they

have integer square roots Similarly, integers such as 1, 8, 27, 64, and 125 are called

perfect cubes because they have integer cube roots.

A common special case of Property 6 is

Using Properties of Radicals

Use the properties of radicals to simplify each expression

Generalizations About nth Roots of Real Numbers

Section P.2 Exponents and Radicals 21

WARNING / CAUTION

When you simplify a radical, it

is important that both expressions

are defined for the same values

of the variable For instance,

are both defined only

for nonnegative values of

Similarly, in Example 10(c),

defined for all real values of x.

1 All possible factors have been removed from the radical.

2 All fractions have radical-free denominators (accomplished by a process called

rationalizing the denominator).

3 The index of the radical is reduced

To simplify a radical, factor the radicand into factors whose exponents are multiples of the index The roots of these factors are written outside the radical, and the

“leftover” factors make up the new radicand

Simplifying Even Roots

Perfect Leftover 4th power factor

a.

Perfect Leftover square factor

b. Find largest square factor.

Find root of perfect square.

c.

Now try Exercise 79(a)

Simplifying Odd Roots

Perfect Leftover cube factor

a.

Perfect Leftover cube factor

b. Find largest cube factor.

Find root of perfect cube.

c. Find largest cube factor.

Find root of perfect cube.

Now try Exercise 79(b)

Radical expressions can be combined (added or subtracted) if they are like radicals—that is, if they have the same index and radicand For instance,

two radicals can be combined, you should first simplify each radical

Combining Radicals

Find square roots and multiply by coefficients.

Combine like terms.

Simplify.

Find cube roots.

Combine like terms.

Now try Exercise 87

Rationalizing Denominators and Numerators

are conjugates of each other If then the rationalizing factor for is itself,For cube roots, choose a rationalizing factor that generates a perfect cube

Rationalizing Single-Term Denominators

Rationalize the denominator of each expression

Section P.2 Exponents and Radicals 23

Definition of Rational Exponents

If a is a real number and n is a positive integer such that the principal nth root of

a exists, then is defined as

where is the rational exponent of a.

Moreover, if m is a positive integer that has no common factor with n, then

Do not confuse the expression

with the expression

In general,

does not equal

Rationalizing a Denominator with Two Terms

Use Distributive Property.

Simplify.

Simplify.

Now try Exercise 97

Sometimes it is necessary to rationalize the numerator of an expression For instance, in Section P.5 you will use the technique shown in the next example torationalize the numerator of an expression from calculus

The numerator of a rational exponent denotes the power to which the base is raised, and the denominator denotes the index or the root to be taken.

When you are working with rational exponents, the properties of integer exponentsstill apply For instance,

Changing From Radical to Exponential Form a.

b.

c.

Now try Exercise 103

Changing From Exponential to Radical Form a.

b.

c.

d.

Now try Exercise 105

Rational exponents are useful for evaluating roots of numbers on a calculator, forreducing the index of a radical, and for simplifying expressions in calculus

Simplifying with Rational Exponents a.

b.

d.

e.

Now try Exercise 115

is not a real number

212 1  0

x12

x12

 2x  1, 2x  14 2x  1  2x  1 4   1 

Rational exponents can be

tricky, and you must remember

that the expression is not

defined unless is a real

number This restriction

produces some unusual-looking

results For instance, the number

There are four methods of

evalu-ating radicals on most graphing

calculators For square roots, you

can use the square root key

For cube roots, you can use the

cube root key For other

roots, you can first convert the

radical to exponential form and

then use the exponential key ,

or you can use the xth root key

(or menu choice) Consult

the user’s guide for your calculator

for specific keystrokes

Power Index



 3

 x

Section P.2 Exponents and Radicals 25

P.2

VOCABULARY: Fill in the blanks

1 In the exponential form is the and is the

2 A convenient way of writing very large or very small numbers is called .

3 One of the two equal factors of a number is called a of the number.

4 The of a number is the th root that has the same sign as and is denoted by

5 In the radical form the positive integer is called the of the radical and the number is called

the

6 When an expression involving radicals has all possible factors removed, radical-free denominators, and a reduced

index, it is in

7 Radical expressions can be combined (added or subtracted) if they are .

9 The process used to create a radical-free denominator is known as the denominator.

10 In the expression denotes the to which the base is raised and denotes the or root

to be taken

SKILLS AND APPLICATIONS

n m

b m ,

a  b m

a b m

a n

n

a,

n

a a,

n a

a n

In Exercises 19 –22, use a calculator to evaluate the

expression (If necessary, round your answer to three decimal

5x4 x2

5z3

x 1 3

In Exercises 45–52, write the number in scientific notation.

49 Land area of Earth: 57,300,000 square miles

50 Light year: 9,460,000,000,000 kilometers

51 Relative density of hydrogen: 0.0000899 gram per

cubic centimeter

52 One micron (millionth of a meter): 0.00003937 inch

In Exercises 53– 60, write the number in decimal notation

57 Interior temperature of the sun:

degrees Celsius

60 Gross domestic product of the United States in 2007:

dollars (Source: U.S Department

In Exercises 63 and 64, use a calculator to evaluate each

expression (Round your answer to three decimal places.)

3

16 27

Section P.2 Exponents and Radicals 27

In Exercises 99 –102, rationalize the numerator of the

expression Then simplify your answer

In Exercises 117 and 118, write each expression as a single

radical Then simplify your answer

the length of the pendulum (in feet) Find the period of

a pendulum whose length is 2 feet

120 EROSION A stream of water moving at the rate of feet

Find the size of the largest particle that can be carried

by a stream flowing at the rate of foot per second

121 MATHEMATICAL MODELING A funnel is filled

with water to a height of centimeters The formula

represents the amount of time (in seconds) that it willtake for the funnel to empty

(a) Use the table feature of a graphing utility to find the

times required for the funnel to empty for water

122 SPEED OF LIGHT The speed of light is approximately

11,180,000 miles per minute The distance from thesun to Earth is approximately 93,000,000 miles Findthe time for light to travel from the sun to Earth

EXPLORATION

TRUE OR FALSE? In Exercises 123 and 124, determine

whether the statement is true or false Justify your answer

128 List all possible digits that occur in the units place

of the square of a positive integer Use that list to

129 THINK ABOUT IT Square the real number

and note that the radical is eliminated from the denominator Is this equivalent to rationalizing thedenominator? Why or why not?

8

2

The symbol indicates an example or exercise that highlights

algebraic techniques specifically used in calculus.

The symbol indicates an exercise or a part of an exercise in which

you are instructed to use a graphing utility.

130 CAPSTONE

(a) Explain how to simplify the expression

or why not?

4

x3

3x3y22

Basic Rules of Algebra< /b>

Let and be real numbers, variables, or algebraic expressions

Commutative Property of Addition:

Commutative... 23

Identifying Rules of Algebra< /b>

Identify the rule of algebra illustrated by the statement

a This statement illustrates... called a number.

7 An algebraic expression is a collection of letters called and real numbers called .

8 The of an algebraic expression are those parts