Dataset Viewer
text
stringlengths 8
3.55k
| source
stringlengths 14
72
|
|---|---|
Algebra and Trigonometry 2e
SENIOR CONTRIBUTING AUTHOR
JAY ABRAMSON, ARIZONA STATE UNIVERSITY
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=2&chunk=0
|
OpenStax
Rice University
6100 Main Street MS-375
Houston, Texas 77005
To learn more about OpenStax, visit https://openstax.org.
Individual print copies and bulk orders can be purchased through our website.
©2021 Rice University. Textbook content produced by OpenStax is licensed under a Creative Commons
Attribution 4.0 International License (CC BY 4.0). Under this license, any user of this textbook or the textbook
contents herein must provide proper attribution as follows:
- If you redistribute this textbook in a digital format (including but not limited to PDF and HTML), then you
must retain on every page the following attribution:
“Access for free at openstax.org.”
- If you redistribute this textbook in a print format, then you must include on every physical page the
following attribution:
“Access for free at openstax.org.”
- If you redistribute part of this textbook, then you must retain in every digital format page view (including
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=3&chunk=1
|
following attribution:
“Access for free at openstax.org.”
- If you redistribute part of this textbook, then you must retain in every digital format page view (including
but not limited to PDF and HTML) and on every physical printed page the following attribution:
“Access for free at openstax.org.”
- If you use this textbook as a bibliographic reference, please include https://openstax.org/details/books/
algebra-and-trigonometry-2e in your citation.
For questions regarding this licensing, please contact [email protected].
Trademarks
The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, OpenStax CNX logo,
OpenStax Tutor name, Openstax Tutor logo, Connexions name, Connexions logo, Rice University name, and
Rice University logo are not subject to the license and may not be reproduced without the prior and express
written consent of Rice University.
HARDCOVER BOOK ISBN-13 978-1-711494-04-3
B&W PAPERBACK BOOK ISBN-13 978-1-711494-03-6
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=3&chunk=2
|
written consent of Rice University.
HARDCOVER BOOK ISBN-13 978-1-711494-04-3
B&W PAPERBACK BOOK ISBN-13 978-1-711494-03-6
DIGITAL VERSION ISBN-13 978-1-951693-40-4
ORIGINAL PUBLICATION YEAR 2021
1 2 3 4 5 6 7 8 9 10 RS 21
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=3&chunk=3
|
OPENSTAX
OpenStax provides free, peer-reviewed, openly licensed textbooks for introductory college and Advanced
Placement® courses and low-cost, personalized courseware that helps students learn. A nonprofit ed tech
initiative based at Rice University, we’re committed to helping students access the tools they need to complete
their courses and meet their educational goals.
RICE UNIVERSITY
OpenStax, OpenStax CNX, and OpenStax Tutor are initiatives of Rice University. As a leading research
university with a distinctive commitment to undergraduate education, Rice University aspires to path-breaking
research, unsurpassed teaching, and contributions to the betterment of our world. It seeks to fulfill this
mission by cultivating a diverse community of learning and discovery that produces leaders across the
spectrum of human endeavor.
PHILANTHROPIC SUPPORT
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=4&chunk=4
|
mission by cultivating a diverse community of learning and discovery that produces leaders across the
spectrum of human endeavor.
PHILANTHROPIC SUPPORT
OpenStax is grateful for the generous philanthropic partners who advance our mission to improve educational
access and learning for everyone. To see the impact of our supporter community and our most updated list of
partners, please visit openstax.org/impact.
Arnold Ventures
Chan Zuckerberg Initiative
Chegg, Inc.
Arthur and Carlyse Ciocca Charitable Foundation
Digital Promise
Ann and John Doerr
Bill & Melinda Gates Foundation
Girard Foundation
Google Inc.
The William and Flora Hewlett Foundation
The Hewlett-Packard Company
Intel Inc.
Rusty and John Jaggers
The Calvin K. Kazanjian Economics Foundation
Charles Koch Foundation
Leon Lowenstein Foundation, Inc.
The Maxfield Foundation
Burt and Deedee McMurtry
Michelson 20MM Foundation
National Science Foundation
The Open Society Foundations
Jumee Yhu and David E. Park III
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=4&chunk=5
|
Leon Lowenstein Foundation, Inc.
The Maxfield Foundation
Burt and Deedee McMurtry
Michelson 20MM Foundation
National Science Foundation
The Open Society Foundations
Jumee Yhu and David E. Park III
Brian D. Patterson USA-International Foundation
The Bill and Stephanie Sick Fund
Steven L. Smith & Diana T. Go
Stand Together
Robin and Sandy Stuart Foundation
The Stuart Family Foundation
Tammy and Guillermo Treviño
Valhalla Charitable Foundation
White Star Education Foundation
Schmidt Futures
William Marsh Rice University
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=4&chunk=6
|
Study where you want, what
you want, when you want.
Access. The future of education.
openstax.org
When you access your book in our web view, you can use our new online
highlighting and note-taking features to create your own study guides.
Our books are free and flexible, forever.
Get started at openstax.org/details/books/algebra-and-trigonometry-2e
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=5&chunk=7
|
Contents
Preface 1
Prerequisites 71
Introduction to Prerequisites 7
1.1 Real Numbers: Algebra Essentials 7
1.2 Exponents and Scientific Notation 24
1.3 Radicals and Rational Exponents 39
1.4 Polynomials 50
1.5 Factoring Polynomials 59
1.6 Rational Expressions 68
Chapter Review 76
Exercises 79
Equations and Inequalities 832
Introduction to Equations and Inequalities 83
2.1 The Rectangular Coordinate Systems and Graphs 83
2.2 Linear Equations in One Variable 98
2.3 Models and Applications 115
2.4 Complex Numbers 125
2.5 Quadratic Equations 134
2.6 Other Types of Equations 149
2.7 Linear Inequalities and Absolute Value Inequalities 161
Chapter Review 172
Exercises 175
Functions 1813
Introduction to Functions 181
3.1 Functions and Function Notation 181
3.2 Domain and Range 205
3.3 Rates of Change and Behavior of Graphs 223
3.4 Composition of Functions 239
3.5 Transformation of Functions 255
3.6 Absolute Value Functions 287
3.7 Inverse Functions 295
Chapter Review 310
Exercises 314
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=6&chunk=8
|
3.4 Composition of Functions 239
3.5 Transformation of Functions 255
3.6 Absolute Value Functions 287
3.7 Inverse Functions 295
Chapter Review 310
Exercises 314
Linear Functions 3234
Introduction to Linear Functions 323
4.1 Linear Functions 323
4.2 Modeling with Linear Functions 360
4.3 Fitting Linear Models to Data 374
Chapter Review 388
Exercises 389
Polynomial and Rational Functions 3995
Introduction to Polynomial and Rational Functions 399
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=6&chunk=9
|
5.1 Quadratic Functions 400
5.2 Power Functions and Polynomial Functions 419
5.3 Graphs of Polynomial Functions 438
5.4 Dividing Polynomials 460
5.5 Zeros of Polynomial Functions 471
5.6 Rational Functions 484
5.7 Inverses and Radical Functions 508
5.8 Modeling Using Variation 521
Chapter Review 531
Exercises 535
Exponential and Logarithmic Functions 5416
Introduction to Exponential and Logarithmic Functions 541
6.1 Exponential Functions 542
6.2 Graphs of Exponential Functions 561
6.3 Logarithmic Functions 575
6.4 Graphs of Logarithmic Functions 584
6.5 Logarithmic Properties 606
6.6 Exponential and Logarithmic Equations 618
6.7 Exponential and Logarithmic Models 631
6.8 Fitting Exponential Models to Data 650
Chapter Review 666
Exercises 671
The Unit Circle: Sine and Cosine Functions 6817
Introduction to The Unit Circle: Sine and Cosine Functions 681
7.1 Angles 682
7.2 Right Triangle Trigonometry 704
7.3 Unit Circle 717
7.4 The Other Trigonometric Functions 736
Chapter Review 751
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=7&chunk=10
|
Introduction to The Unit Circle: Sine and Cosine Functions 681
7.1 Angles 682
7.2 Right Triangle Trigonometry 704
7.3 Unit Circle 717
7.4 The Other Trigonometric Functions 736
Chapter Review 751
Exercises 754
Periodic Functions 7598
Introduction to Periodic Functions 759
8.1 Graphs of the Sine and Cosine Functions 759
8.2 Graphs of the Other Trigonometric Functions 779
8.3 Inverse Trigonometric Functions 800
Chapter Review 813
Exercises 814
Trigonometric Identities and Equations 8219
Introduction to Trigonometric Identities and Equations 821
9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions
821
9.2 Sum and Difference Identities 832
9.3 Double-Angle, Half-Angle, and Reduction Formulas 847
9.4 Sum-to-Product and Product-to-Sum Formulas 860
9.5 Solving Trigonometric Equations 868
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=7&chunk=11
|
Chapter Review 884
Exercises 887
Further Applications of Trigonometry 89310
Introduction to Further Applications of Trigonometry 893
10.1 Non-right Triangles: Law of Sines 893
10.2 Non-right Triangles: Law of Cosines 911
10.3 Polar Coordinates 925
10.4 Polar Coordinates: Graphs 939
10.5 Polar Form of Complex Numbers 958
10.6 Parametric Equations 969
10.7 Parametric Equations: Graphs 982
10.8 Vectors 994
Chapter Review 1016
Exercises 1020
Systems of Equations and Inequalities 102711
Introduction to Systems of Equations and Inequalities 1027
11.1 Systems of Linear Equations: Two Variables 1028
11.2 Systems of Linear Equations: Three Variables 1047
11.3 Systems of Nonlinear Equations and Inequalities: Two Variables 1059
11.4 Partial Fractions 1071
11.5 Matrices and Matrix Operations 1081
11.6 Solving Systems with Gaussian Elimination 1094
11.7 Solving Systems with Inverses 1108
11.8 Solving Systems with Cramer's Rule 1123
Chapter Review 1136
Exercises 1139
Analytic Geometry 114712
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=8&chunk=12
|
11.6 Solving Systems with Gaussian Elimination 1094
11.7 Solving Systems with Inverses 1108
11.8 Solving Systems with Cramer's Rule 1123
Chapter Review 1136
Exercises 1139
Analytic Geometry 114712
Introduction to Analytic Geometry 1147
12.1 The Ellipse 1148
12.2 The Hyperbola 1164
12.3 The Parabola 1181
12.4 Rotation of Axes 1197
12.5 Conic Sections in Polar Coordinates 1212
Chapter Review 1223
Exercises 1225
Sequences, Probability, and Counting Theory 123113
Introduction to Sequences, Probability and Counting Theory 1231
13.1 Sequences and Their Notations 1231
13.2 Arithmetic Sequences 1246
13.3 Geometric Sequences 1258
13.4 Series and Their Notations 1267
13.5 Counting Principles 1281
13.6 Binomial Theorem 1292
13.7 Probability 1299
Chapter Review 1310
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=8&chunk=13
|
Exercises 1313
Proofs, Identities, and Toolkit Functions 1321A
Answer Key 1333
Index 1503
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=9&chunk=14
|
Preface
About OpenStax
OpenStax is part of Rice University, which is a 501(c)(3) nonprofit charitable corporation. As an educational initiative, it's
our mission to transform learning so that education works for every student. Through our partnerships with
philanthropic organizations and our alliance with other educational resource companies, we're breaking down the most
common barriers to learning. Because we believe that everyone should and can have access to knowledge.
About OpenStax Resource
Customization
Algebra and Trigonometry 2e is licensed under a Creative Commons Attribution 4.0 International (CC BY) license, which
means that you can distribute, remix, and build upon the content, as long as you provide attribution to OpenStax and its
content contributors.
Because our books are openly licensed, you are free to use the entire book or pick and choose the sections that are most
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=10&chunk=15
|
content contributors.
Because our books are openly licensed, you are free to use the entire book or pick and choose the sections that are most
relevant to the needs of your course. Feel free to remix the content by assigning your students certain chapters and
sections in your syllabus, in the order that you prefer. You can even provide a direct link in your syllabus to the sections
in the web view of your book.
Instructors also have the option of creating a customized version of their OpenStax book. The custom version can be
made available to students in low-cost print or digital form through their campus bookstore. Visit your book page on
openstax.org for more information.
Art attribution
In Algebra and Trigonometry 2e, most photos and third-party illustrations contain attribution to their creator, rights
holder, host platform, and/or license within the caption. Because the art is openly licensed, anyone may reuse the art as
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=10&chunk=16
|
holder, host platform, and/or license within the caption. Because the art is openly licensed, anyone may reuse the art as
long as they provide the same attribution to its original source. To maximize readability and content flow, mathematical
expressions that are rendered as art do not include attribution in the text. This art can be assumed to be developed by
OpenStax and can be reused under the CC-BY license with attribution.
Errata
All OpenStax textbooks undergo a rigorous review process. However, like any professional-grade textbook, errors
sometimes occur. Since our books are web based, we can make updates periodically when deemed pedagogically
necessary. If you have a correction to suggest, submit it through the link on your book page on openstax.org. Subject
matter experts review all errata suggestions. OpenStax is committed to remaining transparent about all updates, so you
will also find a list of past errata changes on your book page on openstax.org.
Format
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=10&chunk=17
|
will also find a list of past errata changes on your book page on openstax.org.
Format
You can access this textbook for free in web view or PDF through openstax.org, and for a low cost in print.
About Algebra and Trigonometry 2e
Algebra and Trigonometry 2e provides a comprehensive exploration of algebraic principles and meets scope and
sequence requirements for a typical introductory algebra and trigonometry course. The modular approach and the
richness of content ensures that the book meets the needs of a variety of courses. Algebra and Trigonometry 2e offers a
wealth of examples with detailed, conceptual explanations, building a strong foundation in the material before asking
students to apply what they’ve learned.
Coverage and Scope
In determining the concepts, skills, and topics to cover, we engaged dozens of highly experienced instructors with a
range of student audiences. The resulting scope and sequence proceeds logically while allowing for a significant amount
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=10&chunk=18
|
range of student audiences. The resulting scope and sequence proceeds logically while allowing for a significant amount
of flexibility in instruction.
Chapters 1 and 2 provide both a review and foundation for study of functions that begins in Chapter 3. The authors
recognize that while some institutions may find this material a prerequisite, other institutions have told us that they have
a cohort that needs the prerequisite skills built into the course.
• Chapter 1: Prerequisites
• Chapter 2: Equations and Inequalities
Chapters 3-6: The Algebraic Functions
• Chapter 3: Functions
Preface 1
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=10&chunk=19
|
• Chapter 4: Linear Functions
• Chapter 5: Polynomial and Rational Functions
• Chapter 6: Exponential and Logarithm Functions
Chapters 7-10: A Study of Trigonometry
• Chapter 7: The Unit Circle: Sine and Cosine Functions
• Chapter 8: Periodic Functions
• Chapter 9: Trigonometric Identities and Equations
• Chapter 10: Further Applications of Trigonometry
Chapters 11-13: Further Study in Algebra and Trigonometry
• Chapter 11: Systems of Equations and Inequalities
• Chapter 12: Analytic Geometry
• Chapter 13: Sequences, Probability, and Counting Theory
Development Overview
OpenStax Algebra and Trigonometry 2e is the product of a collaborative effort by a group of dedicated authors, editors,
and instructors whose collective passion for this project has resulted in a text that is remarkably unified in purpose and
voice. Special thanks is due to our Lead Author, Jay Abramson of Arizona State University, who provided the overall vision
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=11&chunk=20
|
voice. Special thanks is due to our Lead Author, Jay Abramson of Arizona State University, who provided the overall vision
for the book and oversaw the development of each and every chapter, drawing up the initial blueprint, reading
numerous drafts, and assimilating field reviews into actionable revision plans for our authors and editors.
The collective experience of our author team allowed us to pinpoint the subtopics, exceptions, and individual
connections that give students the most trouble. The textbook is therefore replete with well-designed features and
highlights which help students overcome these barriers. As the students read and practice, they are coached in methods
of thinking through problems and internalizing mathematical processes.
Accuracy of the Content
We understand that precision and accuracy are imperatives in mathematics, and undertook a dedicated accuracy
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=11&chunk=21
|
Accuracy of the Content
We understand that precision and accuracy are imperatives in mathematics, and undertook a dedicated accuracy
program led by experienced faculty.Examples, art, problems, and solutions were reviewed by dedicated faculty, with a
separate team evaluating the answer key and solutions.
The text also benefits from years of usage by thousands of faculty and students. A core aspect of the second edition
revision process included consolidating and ensuring consistency with regard to any errata and corrections that have
been during in the series' extensive usage and incorporation into homework systems.
Changes to the Second Edition
The Algebra and Trigonometry 2e revision focused on mathematical clarity and accuracy as well as inclusivity. Examples,
Exercises, and Solutions were reviewed by multiple faculty experts. All improvement suggestions and errata updates,
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=11&chunk=22
|
Exercises, and Solutions were reviewed by multiple faculty experts. All improvement suggestions and errata updates,
driven by faculty and students from several thousand colleges, were considered and unified across the different formats
of the text.
OpenStax and our authors are aware of the difficulties posed by shifting problem and exercise numbers when textbooks
are revised. In an effort to make the transition to the 2nd edition as seamless as possible, we have minimized any
shifting of exercise numbers.
The revision also focused on supporting inclusive and welcoming learning experiences. The introductory narratives,
example and problem contexts, and even many of the names used for fictional people in the text were all reviewed using
a diversity, equity, and inclusion framework. Several hundred resulting revisions improve the balance and relevance to
the students using the text, while maintaining a variety of applications to diverse careers and academic fields. In
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=11&chunk=23
|
the students using the text, while maintaining a variety of applications to diverse careers and academic fields. In
particular, explanations of scientific and historical aspects of mathematics have been expanded to include more
contributors. For example, the authors added additional historical and multicultural context regarding what is widely
known as Pascal’s Triangle, and similarly added details regarding the international process of decoding the Enigma
machine (including the role of Polish college students). Several chapter opening narratives and in-chapter references are
completely new, and contexts across all chapters were specifically reviewed for equity in gender representation and
connotation.
Finally, prior to the release of this edition, OpenStax published a series of Corequisite Skillsheets to support different
models and approaches to instruction. These remain available, and are described in more detail below.
2 Preface
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=11&chunk=24
|
Pedagogical Foundations and Features
Learning Objectives
Each chapter is divided into multiple sections (or modules), each of which is organized around a set of learning
objectives. The learning objectives are listed explicitly at the beginning of each section and are the focal point of every
instructional element
Narrative text
Narrative text is used to introduce key concepts, terms, and definitions, to provide real-world context, and to provide
transitions between topics and examples. Throughout this book, we rely on a few basic conventions to highlight the
most important ideas:
• Key terms are boldfaced, typically when first introduced and/or when formally defined.
• Key concepts and definitions are called out in a blue box for easy reference.
Examples
Each learning objective is supported by one or more worked examples, that demonstrate the problem-solving
approaches that students must master. The multiple Examples model different approaches to the same type of problem,
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=12&chunk=25
|
approaches that students must master. The multiple Examples model different approaches to the same type of problem,
or introduce similar problems of increasing complexity.
All Examples follow a simple two- or three-part format. The question clearly lays out a mathematical problem to solve.
The Solution walks through the steps, usually providing context for the approach—in other words, why the instructor is
solving the problem in a specific manner. Finally, the Analysis (for select Examples) reflects on the broader implications
of the Solution just shown. Examples are followed by a Try It question, as explained below.
Figures
Algebra and Trigonometry 2e contains more than 2000 figures and illustrations, the vast majority of which are graphs
and diagrams. Art throughout the text adheres to a clear, understated style, drawing the eye to the most important
information in each figure while minimizing visual distractions. Color contrast is employed with discretion to distinguish
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=12&chunk=26
|
information in each figure while minimizing visual distractions. Color contrast is employed with discretion to distinguish
between the different functions or features of a graph.
Supporting Features
Several elements contribute to and check understanding.
• A “How To” is a list of steps necessary to solve a certain type of problem. A How To typically precedes an Example
that proceeds to demonstrate the steps in action.
• A “Try It” exercise immediately follows an Example or a set of related Examples, providing the student with an
immediate opportunity to solve a similar problem. In the PDF and the Web View version of the text, answers to the
Try It exercises are located in the Answer Key.
• A Q&A may appear at any point in the narrative, but most often follows an Example. This feature pre-empts
misconceptions by posing a commonly asked yes/no question, followed by a detailed answer and explanation.
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=12&chunk=27
|
misconceptions by posing a commonly asked yes/no question, followed by a detailed answer and explanation.
• The “Media” icon appears at the conclusion of each section, just prior to the Section Exercises. This icon marks a list
Preface 3
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=12&chunk=28
|
of links to online video tutorials that reinforce the concepts and skills introduced in the section.
While we have selected tutorials that closely align to our learning objectives, we did not produce these tutorials, nor
were they specifically produced or tailored to accompany Algebra and Trigonometry 2e.
Section Exercises
Each section of every chapter concludes with a well-rounded set of exercises that can be assigned as homework or used
selectively for guided practice. With over 6300 exercises across the 13 chapters, instructors should have plenty from
which to choose.
Section Exercises are organized by question type, and generally appear in the following order:
• Verbal questions assess conceptual understanding of key terms and concepts.
• Algebraic problems require students to apply algebraic manipulations demonstrated in the section.
• Graphical problems assess students’ ability to interpret or produce a graph.
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=13&chunk=29
|
• Algebraic problems require students to apply algebraic manipulations demonstrated in the section.
• Graphical problems assess students’ ability to interpret or produce a graph.
• Numeric problems require the student to perform calculations or computations.
• Technology problems encourage exploration through use of a graphing utility, either to visualize or verify algebraic
results or to solve problems via an alternative to the methods demonstrated in the section.
• Extensions pose problems more challenging than the Examples demonstrated in the section. They require students
to synthesize multiple learning objectives or apply critical thinking to solve complex problems.
• Real-World Applications present realistic problem scenarios from fields such as physics, geology, biology, finance,
and the social sciences.
Chapter Review Features
Each chapter concludes with a review of the most important takeaways, as well as additional practice problems that
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=13&chunk=30
|
and the social sciences.
Chapter Review Features
Each chapter concludes with a review of the most important takeaways, as well as additional practice problems that
students can use to prepare for exams.
• Key Terms provides a formal definition for each bold-faced term in the chapter.
• Key Equations presents a compilation of formulas, theorems, and standard-form equations.
• Key Concepts summarizes the most important ideas introduced in each section, linking back to the relevant
Example(s) in case students need to review.
• Chapter Review Exercises include 40-80 practice problems that recall the most important concepts from each
section.
• Practice Test includes 25-50 problems assessing the most important learning objectives from the chapter. Note
that the practice test is not organized by section, and may be more heavily weighted toward cumulative objectives
as opposed to the foundational objectives covered in the opening sections.
Corequisite Support
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=13&chunk=31
|
as opposed to the foundational objectives covered in the opening sections.
Corequisite Support
Each Algebra and Trigonometry 2e section is paired with a thoughtfully developed, topically aligned skills module that
prepares students for the course material. Sharon North (St. Louis Community College) developed a coordinated set of
support resources, which provide review, instruction, and practice for algebra students. The author team identified
foundational skills and concepts, then mapped them to each module. The corequisite sections include conceptual
overviews, worked examples, and guided practice; they incorporate relevant material from OpenStax’s Developmental
Math series. They are available as separate, openly accessible downloads from the student and instructor resources
pages accompanying the text.
Additional Resources
Student and Instructor Resources
We’ve compiled additional resources for both students and instructors, including Getting Started Guides, instructor
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=13&chunk=32
|
pages accompanying the text.
Additional Resources
Student and Instructor Resources
We’ve compiled additional resources for both students and instructors, including Getting Started Guides, instructor
solution manual, Corequisite skillsheets, and PowerPoint slides. Instructor resources require a verified instructor
account, which can be requested on your openstax.org log-in. Take advantage of these resources to supplement your
OpenStax book.
Community Hubs
OpenStax partners with the Institute for the Study of Knowledge Management in Education (ISKME) to offer Community
Hubs on OER Commons—a platform for instructors to share community-created resources that support OpenStax
books, free of charge. Through our Community Hubs, instructors can upload their own materials or download resources
to use in their own courses, including additional ancillaries, teaching material, multimedia, and relevant course content.
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=13&chunk=33
|
to use in their own courses, including additional ancillaries, teaching material, multimedia, and relevant course content.
We encourage instructors to join the hubs for the subjects most relevant to your teaching and research as an
4 Preface
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=13&chunk=34
|
opportunity both to enrich your courses and to engage with other faculty. To reach the Community Hubs, visit
www.oercommons.org/hubs/openstax.
Technology partners
As allies in making high-quality learning materials accessible, our technology partners offer optional low-cost tools that
are integrated with OpenStax books. To access the technology options for your text, visit your book page on
openstax.org.
About the Authors
Senior Contributing Author
Jay Abramson, Arizona State University
Jay Abramson has been teaching Precalculus for over 35 years, the last 20 at Arizona State University, where he is a
principal lecturer in the School of Mathematics and Statistics. His accomplishments at ASU include co-developing the
university’s first hybrid and online math courses as well as an extensive library of video lectures and tutorials. In
addition, he has served as a contributing author for two of Pearson Education’s math programs, NovaNet Precalculus
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=14&chunk=35
|
addition, he has served as a contributing author for two of Pearson Education’s math programs, NovaNet Precalculus
and Trigonometry. Prior to coming to ASU, Jay taught at Texas State Technical College and Amarillo College. He received
Teacher of the Year awards at both institutions.
Contributing Authors
Valeree Falduto, Palm Beach State College
Rachael Gross, Towson University
David Lippman, Pierce College
Melonie Rasmussen, Pierce College
Rick Norwood, East Tennessee State University
Nicholas Belloit, Florida State College Jacksonville
Jean-Marie Magnier, Springfield Technical Community College
Harold Whipple
Christina Fernandez
Reviewers
Phil Clark, Scottsdale Community College
Michael Cohen, Hofstra University
Charles Conrad, Volunteer State Community College
David French, Tidewater Community College
Matthew Goodell, SUNY Ulster
Lance Hemlow, Raritan Valley Community College
Dongrin Kim, Arizona State University
Cynthia Landrigan, Eerie Community College
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=14&chunk=36
|
David French, Tidewater Community College
Matthew Goodell, SUNY Ulster
Lance Hemlow, Raritan Valley Community College
Dongrin Kim, Arizona State University
Cynthia Landrigan, Eerie Community College
Wendy Lightheart, Lane Community College
Chinenye Ofodile, Albany State University
Carl Penziul, Tompkins-Cortland Community College
Sandra Nite, Texas A&M University
Eugenia Peterson, Richard J. Daley College
Rhonda Porter, Albany State University
Michael Price, University of Oregon
Steven Purtee, Valencia College
William Radulovich, Florida State College Jacksonville
Camelia Salajean, City Colleges of Chicago
Katy Shields, Oakland Community College
Nathan Schrenk, ECPI University
Pablo Suarez, Delaware State University
Allen Wolmer, Atlanta Jewish Academy
The following faculty contributed to the development of OpenStax
Precalculus, the text from which this product was
updated and derived.
Precalculus Reviewers
Nina Alketa, Cecil College
Kiran Bhutani, Catholic University of America
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=14&chunk=37
|
Precalculus, the text from which this product was
updated and derived.
Precalculus Reviewers
Nina Alketa, Cecil College
Kiran Bhutani, Catholic University of America
Brandie Biddy, Cecil College
Lisa Blank, Lyme Central School
Preface 5
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=14&chunk=38
|
Bryan Blount, Kentucky Wesleyan College
Jessica Bolz, The Bryn Mawr School
Sheri Boyd, Rollins College
Sarah Brewer, Alabama School of Math and Science
Charles Buckley, St. Gregory's University
Michael Cohen, Hofstra University
Kenneth Crane, Texarkana College
Rachel Cywinski, Alamo Colleges
Nathan Czuba
Srabasti Dutta, Ashford University
Kristy Erickson, Cecil College
Nicole Fernandez, Georgetown University / Kent State University
David French, Tidewater Community College
Douglas Furman, SUNY Ulster
Lance Hemlow, Raritan Valley Community College
Erinn Izzo, Nicaragua Christian Academy
John Jaffe
Jerry Jared, Blue Ridge School
Stan Kopec, Mount Wachusett Community College
Kathy Kovacs
Cynthia Landrigan, Erie Community College
Sara Lenhart, Christopher Newport University
Wendy Lightheart, Lane Community College
Joanne Manville, Bunker Hill Community College
Karla McCavit, Albion College
Cynthia McGinnis, Northwest Florida State College
Lana Neal, University of Texas at Austin
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=15&chunk=39
|
Joanne Manville, Bunker Hill Community College
Karla McCavit, Albion College
Cynthia McGinnis, Northwest Florida State College
Lana Neal, University of Texas at Austin
Rhonda Porter, Albany State University
Steven Purtee, Valencia College
William Radulovich, Florida State College Jacksonville
Alice Ramos, Bethel College
Nick Reynolds, Montgomery Community College
Amanda Ross, A. A. Ross Consulting and Research, LLC
Erica Rutter, Arizona State University
Sutandra Sarkar, Georgia State University
Willy Schild, Wentworth Institute of Technology
Todd Stephen, Cleveland State University
Scott Sykes, University of West Georgia
Linda Tansil, Southeast Missouri State University
John Thomas, College of Lake County
Diane Valade, Piedmont Virginia Community College
Allen Wolmer, Atlanta Jewish Academy
6 Preface
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=15&chunk=40
|
Credit: Andreas Kambanls
Chapter Outline
1.1 Real Numbers: Algebra Essentials
1.2 Exponents and Scientific Notation
1.3 Radicals and Rational Exponents
1.4 Polynomials
1.5 Factoring Polynomials
1.6 Rational Expressions
Introduction to Prerequisites
It’s a cold day in Antarctica. In fact, it’s always a cold day in Antarctica. Earth’s southernmost continent, Antarctica
experiences the coldest, driest, and windiest conditions known. The coldest temperature ever recorded, over one
hundred degrees below zero on the Celsius scale, was recorded by remote satellite. It is no surprise then, that no native
human population can survive the harsh conditions. Only explorers and scientists brave the environment for any length
of time.
Measuring and recording the characteristics of weather conditions in Antarctica requires a use of different kinds of
numbers. For tens of thousands of years, humans have undertaken methods to tally, track, and record numerical
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=16&chunk=41
|
numbers. For tens of thousands of years, humans have undertaken methods to tally, track, and record numerical
information. While we don't know much about their usage, the Lebombo Bone (dated to about 35,000 BCE) and the
Ishango Bone (dated to about 20,000 BCE) are among the earliest mathematical artifacts. Found in Africa, their clearly
deliberate groupings of notches may have been used to track time, moon cycles, or other information. Performing
calculations with them and using the results to make predictions requires an understanding of relationships among
numbers. In this chapter, we will review sets of numbers and properties of operations used to manipulate numbers. This
understanding will serve as prerequisite knowledge throughout our study of algebra and trigonometry.
1.1 Real Numbers: Algebra Essentials
Learning Objectives
In this section, you will:
Classify a real number as a natural, whole, integer, rational, or irrational number.
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=16&chunk=42
|
1.1 Real Numbers: Algebra Essentials
Learning Objectives
In this section, you will:
Classify a real number as a natural, whole, integer, rational, or irrational number.
Perform calculations using order of operations.
Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity.
Evaluate algebraic expressions.
Simplify algebraic expressions.
PREREQUISITES
1
1 • Introduction 7
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=16&chunk=43
|
It is often said that mathematics is the language of science. If this is true, then an essential part of the language of
mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or
enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity—a
sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to
improved communications and the spread of civilization.
Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they
used them to represent the amount when a quantity was divided into equal parts.
But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the
existence of nothing? From earliest times, people had thought of a “base state” while counting and used various
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=17&chunk=44
|
existence of nothing? From earliest times, people had thought of a “base state” while counting and used various
symbols to represent this null condition. However, it was not until about the fifth century CE in India that zero was added
to the number system and used as a numeral in calculations.
Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative
numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting
numbers expanded the number system even further.
Because of the evolution of the number system, we can now perform complex calculations using these and other
categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers,
and the use of numbers in expressions.
Classifying a Real Number
The numbers we use for counting, or enumerating items, are the natural numbers: 1, 2, 3, 4, 5, and so on. We describe
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=17&chunk=45
|
and the use of numbers in expressions.
Classifying a Real Number
The numbers we use for counting, or enumerating items, are the natural numbers: 1, 2, 3, 4, 5, and so on. We describe
them in set notation as where the ellipsis (…) indicates that the numbers continue to infinity. The natural
numbers are, of course, also called the counting numbers. Any time we enumerate the members of a team, count the
coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is
the set of natural numbers plus zero:
The set of integers adds the opposites of the natural numbers to the set of whole numbers:
It is useful to note that the set of integers is made up of three distinct subsets: negative
integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to
think about it is that the natural numbers are a subset of the integers.
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=17&chunk=46
|
integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to
think about it is that the natural numbers are a subset of the integers.
The set of rational numbers is written as Notice from the definition that rational
numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the
denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number
with a denominator of 1.
Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be
represented as either:
ⓐ a terminating decimal: or ⓑ a repeating decimal:
We use a line drawn over the repeating block of numbers instead of writing the group multiple times.
EXAMPLE 1
Writing Integers as Rational Numbers
Write each of the following as a rational number.
ⓐ 7 ⓑ 0 ⓒ –8
Solution
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=17&chunk=47
|
EXAMPLE 1
Writing Integers as Rational Numbers
Write each of the following as a rational number.
ⓐ 7 ⓑ 0 ⓒ –8
Solution
Write a fraction with the integer in the numerator and 1 in the denominator.
ⓐ ⓑ ⓒ
TRY IT #1 Write each of the following as a rational number.
ⓐ 11 ⓑ 3 ⓒ –4
8 1 • Prerequisites
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=17&chunk=48
|
EXAMPLE 2
Identifying Rational Numbers
Write each of the following rational numbers as either a terminating or repeating decimal.
ⓐ ⓑ ⓒ
Solution
Write each fraction as a decimal by dividing the numerator by the denominator.
ⓐ a repeating decimal ⓑ (or 3.0), a terminating decimal
ⓒ a terminating decimal
TRY IT #2 Write each of the following rational numbers as either a terminating or repeating decimal.
ⓐ ⓑ ⓒ
Irrational Numbers
At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for
instance, may have found that the diagonal of a square with unit sides was not 2 or even but was something else. Or
a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit
more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=18&chunk=49
|
more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as
fractions. These numbers make up the set of irrational numbers. Irrational numbers cannot be expressed as a fraction
of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if
it is not rational. So we write this as shown.
EXAMPLE 3
Differentiating Rational and Irrational Numbers
Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a
terminating or repeating decimal.
ⓐ ⓑ ⓒ ⓓ ⓔ
Solution
ⓐ This can be simplified as Therefore, is rational.
ⓑ Because it is a fraction of integers, is a rational number. Next, simplify and divide.
So, is rational and a repeating decimal.
ⓒ This cannot be simplified any further. Therefore, is an irrational number.
ⓓ Because it is a fraction of integers, is a rational number. Simplify and divide.
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=18&chunk=50
|
ⓒ This cannot be simplified any further. Therefore, is an irrational number.
ⓓ Because it is a fraction of integers, is a rational number. Simplify and divide.
So, is rational and a terminating decimal.
ⓔ is not a terminating decimal. Also note that there is no repeating pattern because the group
of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational
number. It is an irrational number.
1.1 • Real Numbers: Algebra Essentials 9
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=18&chunk=51
|
TRY IT #3 Determine whether each of the following numbers is rational or irrational. If it is rational,
determine whether it is a terminating or repeating decimal.
ⓐ ⓑ ⓒ ⓓ ⓔ
Real Numbers
Given any number n, we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational
numbers together make up the set of real numbers. As we saw with integers, the real numbers can be divided into
three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and
irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.
The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative
numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=19&chunk=52
|
numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer
(or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The
converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-
to-one correspondence. We refer to this as the real number line as shown in Figure 1.
Figure 1 The real number line
EXAMPLE 4
Classifying Real Numbers
Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or
the right of 0 on the number line?
ⓐ ⓑ ⓒ ⓓ ⓔ
Solution
ⓐ is negative and rational. It lies to the left of 0 on the number line.
ⓑ is positive and irrational. It lies to the right of 0.
ⓒ is negative and rational. It lies to the left of 0.
ⓓ is negative and irrational. It lies to the left of 0.
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=19&chunk=53
|
ⓑ is positive and irrational. It lies to the right of 0.
ⓒ is negative and rational. It lies to the left of 0.
ⓓ is negative and irrational. It lies to the left of 0.
ⓔ is a repeating decimal so it is rational and positive. It lies to the right of 0.
TRY IT #4 Classify each number as either positive or negative and as either rational or irrational. Does the
number lie to the left or the right of 0 on the number line?
ⓐ ⓑ ⓒ ⓓ ⓔ
Sets of Numbers as Subsets
Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset
relationship between the sets of numbers we have encountered so far. These relationships become more obvious when
seen as a diagram, such as Figure 2.
10 1 • Prerequisites
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=19&chunk=54
|
Figure 2 Sets of numbers
N: the set of natural numbers
W: the set of whole numbers
I: the set of integers
Q: the set of rational numbers
Q´: the set of irrational numbers
Sets of Numbers
The set of natural numbers includes the numbers used for counting:
The set of whole numbers is the set of natural numbers plus zero:
The set of integers adds the negative natural numbers to the set of whole numbers:
The set of rational numbers includes fractions written as
The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating:
EXAMPLE 5
Differentiating the Sets of Numbers
Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or
irrational number (Q′).
ⓐ ⓑ ⓒ ⓓ ⓔ
Solution
N W I Q Q′
a. X X X X
b. X
c. X
d. –6 X X
e. 3.2121121112... X
1.1 • Real Numbers: Algebra Essentials 11
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=20&chunk=55
|
TRY IT #5 Classify each number as being a natural number (N), whole number (W), integer (I), rational
number (Q), and/or irrational number (Q′).
ⓐ ⓑ ⓒ ⓓ ⓔ
Performing Calculations Using the Order of Operations
When we multiply a number by itself, we square it or raise it to a power of 2. For example, We can raise
any number to any power. In general, the exponential notation means that the number or variable is used as a
factor times.
In this notation, is read as the nth power of or to the where is called the base and is called the exponent. A
term in exponential notation may be part of a mathematical expression, which is a combination of numbers and
operations. For example, is a mathematical expression.
To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any
random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=21&chunk=56
|
random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.
Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that
anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars
are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within
grouping symbols.
The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right
and finally addition and subtraction from left to right.
Let’s take a look at the expression provided.
There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so
simplify as 16.
Next, perform multiplication or division, left to right.
Lastly, perform addition or subtraction, left to right.
Therefore,
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=21&chunk=57
|
simplify as 16.
Next, perform multiplication or division, left to right.
Lastly, perform addition or subtraction, left to right.
Therefore,
For some complicated expressions, several passes through the order of operations will be needed. For instance, there
may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following
the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.
Order of Operations
Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the
12 1 • Prerequisites
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=21&chunk=58
|
...
acronym PEMDAS:
P(arentheses)
E(xponents)
M(ultiplication) and D(ivision)
A(ddition) and S(ubtraction)
HOW TO
Given a mathematical expression, simplify it using the order of operations.
Step 1. Simplify any expressions within grouping symbols.
Step 2. Simplify any expressions containing exponents or radicals.
Step 3. Perform any multiplication and division in order, from left to right.
Step 4. Perform any addition and subtraction in order, from left to right.
EXAMPLE 6
Using the Order of Operations
Use the order of operations to evaluate each of the following expressions.
ⓐ ⓑ ⓒ ⓓ
ⓔ Solution
ⓐ
ⓑ
Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the
fraction bar is considered a grouping symbol so the numerator is considered to be grouped.
ⓒ
1.1 • Real Numbers: Algebra Essentials 13
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=22&chunk=59
|
ⓓ
In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the
last step.
ⓔ
TRY IT #6 Use the order of operations to evaluate each of the following expressions.
ⓐ ⓑ ⓒ
ⓓ ⓔ
Using Properties of Real Numbers
For some activities we perform, the order of certain operations does not matter, but the order of other operations does.
For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does
matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.
Commutative Properties
The commutative property of addition states that numbers may be added in any order without affecting the sum.
We can better see this relationship when using real numbers.
Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without
affecting the product.
Again, consider an example with real numbers.
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=23&chunk=60
|
Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without
affecting the product.
Again, consider an example with real numbers.
It is important to note that neither subtraction nor division is commutative. For example, is not the same as
Similarly,
Associative Properties
The associative property of multiplication tells us that it does not matter how we group numbers when multiplying.
We can move the grouping symbols to make the calculation easier, and the product remains the same.
Consider this example.
The associative property of addition tells us that numbers may be grouped differently without affecting the sum.
This property can be especially helpful when dealing with negative integers. Consider this example.
14 1 • Prerequisites
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=23&chunk=61
|
Are subtraction and division associative? Review these examples.
As we can see, neither subtraction nor division is associative.
Distributive Property
The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the
sum.
This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.
Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and
adding the products.
To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not
true, as we can see in this example.
A special case of the distributive property occurs when a sum of terms is subtracted.
For example, consider the difference We can rewrite the difference of the two terms 12 and by
turning the subtraction expression into addition of the opposite. So instead of subtracting we add the opposite.
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=24&chunk=62
|
turning the subtraction expression into addition of the opposite. So instead of subtracting we add the opposite.
Now, distribute and simplify the result.
This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce
algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can
rewrite the last example.
Identity Properties
The identity property of addition states that there is a unique number, called the additive identity (0) that, when added
to a number, results in the original number.
The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that,
when multiplied by a number, results in the original number.
For example, we have and There are no exceptions for these properties; they work for every
real number, including 0 and 1.
1.1 • Real Numbers: Algebra Essentials 15
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=24&chunk=63
|
Inverse Properties
The inverse property of addition states that, for every real number a, there is a unique number, called the additive
inverse (or opposite), denoted by (−a), that, when added to the original number, results in the additive identity, 0.
For example, if the additive inverse is 8, since
The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined.
The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or
reciprocal), denoted that, when multiplied by the original number, results in the multiplicative identity, 1.
For example, if the reciprocal, denoted is because
Properties of Real Numbers
The following properties hold for real numbers a, b, and c.
Addition Multiplication
Commutative
Property
Associative
Property
Distributive
Property
Identity
Property
There exists a unique real number called the
additive identity, 0, such that, for any real
number a
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=25&chunk=64
|
Commutative
Property
Associative
Property
Distributive
Property
Identity
Property
There exists a unique real number called the
additive identity, 0, such that, for any real
number a
There exists a unique real number called the
multiplicative identity, 1, such that, for any real
number a
Inverse
Property
Every real number a has an additive inverse,
or opposite, denoted –a, such that
Every nonzero real number a has a
multiplicative inverse, or reciprocal, denoted
such that
EXAMPLE 7
Using Properties of Real Numbers
Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.
ⓐ ⓑ ⓒ ⓓ ⓔ
16 1 • Prerequisites
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=25&chunk=65
|
Solution
ⓐ
ⓑ
ⓒ
ⓓ
ⓔ
TRY IT #7 Use the properties of real numbers to rewrite and simplify each expression. State which
properties apply.
ⓐ ⓑ ⓒ
ⓓ ⓔ
Evaluating Algebraic Expressions
So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see
expressions such as or In the expression 5 is called a constant because it does not vary
and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the
variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations
of addition, subtraction, multiplication, and division.
We have already seen some real number examples of exponential notation, a shorthand method of writing products of
the same factor. When variables are used, the constants and variables are treated the same way.
In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=26&chunk=66
|
In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or
variables.
Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the
algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a
given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify
the resulting expression using the order of operations. If the algebraic expression contains more than one variable,
replace each variable with its assigned value and simplify the expression as before.
1.1 • Real Numbers: Algebra Essentials 17
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=26&chunk=67
|
EXAMPLE 8
Describing Algebraic Expressions
List the constants and variables for each algebraic expression.
ⓐ x + 5 ⓑ ⓒ
Solution
Constants Variables
a. x + 5 5 x
b.
c. 2
TRY IT #8 List the constants and variables for each algebraic expression.
ⓐ ⓑ 2(L + W) ⓒ
EXAMPLE 9
Evaluating an Algebraic Expression at Different Values
Evaluate the expression for each value for x.
ⓐ ⓑ ⓒ ⓓSolution
ⓐ Substitute 0 for ⓑ Substitute 1 for ⓒ Substitute for ⓓ Substitute for
TRY IT #9 Evaluate the expression for each value for y.
ⓐ ⓑ ⓒ ⓓ
EXAMPLE 10
Evaluating Algebraic Expressions
Evaluate each expression for the given values.
ⓐ for ⓑ for ⓒ for ⓓ for
ⓔ for
Solution
ⓐ Substitute for ⓑ Substitute 10 for ⓒ Substitute 5 for
18 1 • Prerequisites
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=27&chunk=68
|
ⓓ Substitute 11 for and –8 for ⓔ Substitute 2 for and 3 for
TRY IT #10 Evaluate each expression for the given values.
ⓐ for ⓑ for ⓒ for
ⓓ for ⓔ for
Formulas
An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical
or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true,
the solutions, are found using the properties of real numbers and other results. For example, the equation
has the solution of 3 because when we substitute 3 for in the equation, we obtain the true statement
A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is
a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the
most common examples is the formula for finding the area of a circle in terms of the radius of the circle:
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=28&chunk=69
|
most common examples is the formula for finding the area of a circle in terms of the radius of the circle:
For any value of the area can be found by evaluating the expression
EXAMPLE 11
Using a Formula
A right circular cylinder with radius and height has the surface area (in square units) given by the formula
See Figure 3. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in
terms of
Figure 3 Right circular cylinder
Solution
Evaluate the expression for and
The surface area is square inches.
TRY IT #11 A photograph with length L and width W is placed in a mat of width 8 centimeters (cm). The area
of the mat (in square centimeters, or cm2) is found to be See
Figure 4. Find the area of a mat for a photograph with length 32 cm and width 24 cm.
1.1 • Real Numbers: Algebra Essentials 19
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=28&chunk=70
|
Figure 4
Simplifying Algebraic Expressions
Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so,
we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic
expressions.
EXAMPLE 12
Simplifying Algebraic Expressions
Simplify each algebraic expression.
ⓐ ⓑ ⓒ ⓓ
Solution
ⓐ
ⓑ
ⓒ
ⓓ
TRY IT #12 Simplify each algebraic expression.
ⓐ ⓑ ⓒ
ⓓ
EXAMPLE 13
Simplifying a Formula
A rectangle with length and width has a perimeter given by Simplify this expression.
20 1 • Prerequisites
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=29&chunk=71
|
Solution
TRY IT #13 If the amount is deposited into an account paying simple interest for time the total value of
the deposit is given by Simplify the expression. (This formula will be explored in
more detail later in the course.)
MEDIA
Access these online resources for additional instruction and practice with real numbers.
Simplify an Expression. (http://openstax.org/l/simexpress)
Evaluate an Expression 1. (http://openstax.org/l/ordofoper1)
Evaluate an Expression 2. (http://openstax.org/l/ordofoper2)
1.1 SECTION EXERCISES
Verbal
1. Is an example of a
rational terminating,
rational repeating, or
irrational number? Tell why
it fits that category.
2. What is the order of
operations? What acronym
is used to describe the order
of operations, and what
does it stand for?
3. What do the Associative
Properties allow us to do
when following the order of
operations? Explain your
answer.
Numeric
For the following exercises, simplify the given expression.
4. 5. 6.
7. 8. 9.
10. 11. 12.
13. 14. 15.
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=30&chunk=72
|
Properties allow us to do
when following the order of
operations? Explain your
answer.
Numeric
For the following exercises, simplify the given expression.
4. 5. 6.
7. 8. 9.
10. 11. 12.
13. 14. 15.
16. 17. 18.
19. 20. 21.
22. 23. 24.
25. 26. 27.
1.1 • Real Numbers: Algebra Essentials 21
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=30&chunk=73
|
Algebraic
For the following exercises, evaluate the expressions using the given variable.
28. for 29. for 30. for
31. for 32. for 33. for
34. For the
for
35. for 36. for
37. for
For the following exercises, simplify the expression.
38. 39. 40.
41. 42. 43.
44. 45. 46.
47. 48. 49.
50. 51. 52.
Real-World Applications
For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a
streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor’s dog.
53. Write the expression that represents the number
of dollars Fred keeps (and does not put in his
savings account). Remember the order of
operations.
54. How much money does Fred keep?
For the following exercises, solve the given problem.
55. According to the U.S. Mint, the diameter of a
quarter is 0.955 inches. The circumference of the
quarter would be the diameter multiplied by Is
the circumference of a quarter a whole number, a
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=31&chunk=74
|
55. According to the U.S. Mint, the diameter of a
quarter is 0.955 inches. The circumference of the
quarter would be the diameter multiplied by Is
the circumference of a quarter a whole number, a
rational number, or an irrational number?
56. Jessica and her roommate, Adriana, have decided
to share a change jar for joint expenses. Jessica
put her loose change in the jar first, and then
Adriana put her change in the jar. We know that it
does not matter in which order the change was
added to the jar. What property of addition
describes this fact?
22 1 • Prerequisites
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=31&chunk=75
|
For the following exercises, consider this scenario: There is a mound of pounds of gravel in a quarry. Throughout the
day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the
mound. At the end of the day, the mound has 1,200 pounds of gravel.
57. Write the equation that describes the situation. 58. Solve for g.
For the following exercise, solve the given problem.
59. Ramon runs the marketing department at their
company. Their department gets a budget every
year, and every year, they must spend the entire
budget without going over. If they spend less than
the budget, then the department gets a smaller
budget the following year. At the beginning of this
year, Ramon got $2.5 million for the annual
marketing budget. They must spend the budget
such that What property of
addition tells us what the value of x must be?
Technology
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=32&chunk=76
|
year, Ramon got $2.5 million for the annual
marketing budget. They must spend the budget
such that What property of
addition tells us what the value of x must be?
Technology
For the following exercises, use a graphing calculator to solve for x. Round the answers to the nearest hundredth.
60. 61.
Extensions
62. If a whole number is not a
natural number, what must
the number be?
63. Determine whether the
statement is true or false:
The multiplicative inverse
of a rational number is also
rational.
64. Determine whether the
statement is true or false:
The product of a rational
and irrational number is
always irrational.
65. Determine whether the
simplified expression is
rational or irrational:
66. Determine whether the
simplified expression is
rational or irrational:
67. The division of two natural
numbers will always result
in what type of number?
68. What property of real
numbers would simplify
the following expression:
1.1 • Real Numbers: Algebra Essentials 23
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=32&chunk=77
|
1.2 Exponents and Scientific Notation
Learning Objectives
In this section, you will:
Use the product rule of exponents.
Use the quotient rule of exponents.
Use the power rule of exponents.
Use the zero exponent rule of exponents.
Use the negative rule of exponents.
Find the power of a product and a quotient.
Simplify exponential expressions.
Use scientific notation.
Mathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be
obvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be
perceived and recorded by a digital camera. A particular camera might record an image that is 2,048 pixels by 1,536
pixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to 48 bits per
frame, and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=33&chunk=78
|
frame, and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information
used to film a one-hour (3,600-second) digital film is then an extremely large number.
Using a calculator, we enter and press ENTER. The calculator displays 1.304596316E13.
What does this mean? The “E13” portion of the result represents the exponent 13 of ten, so there are a maximum of
approximately bits of data in that one-hour film. In this section, we review rules of exponents first and then
apply them to calculations involving very large or small numbers.
Using the Product Rule of Exponents
Consider the product Both terms have the same base, x, but they are raised to different exponents. Expand each
expression, and then rewrite the resulting expression.
The result is that
Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=33&chunk=79
|
expression, and then rewrite the resulting expression.
The result is that
Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying
exponential expressions with the same base, we write the result with the common base and add the exponents. This is
the product rule of exponents.
Now consider an example with real numbers.
We can always check that this is true by simplifying each exponential expression. We find that is 8, is 16, and is
128. The product equals 128, so the relationship is true. We can use the product rule of exponents to simplify
expressions that are a product of two numbers or expressions with the same base but different exponents.
The Product Rule of Exponents
For any real number and natural numbers and the product rule of exponents states that
EXAMPLE 1
Using the Product Rule
Write each of the following products with a single base. Do not simplify further.
24 1 • Prerequisites
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=33&chunk=80
|
ⓐ ⓑ ⓒSolution
Use the product rule to simplify each expression.
ⓐ ⓑ ⓒAt first, it may appear that we cannot simplify a product of three factors. However, using the associative property of
multiplication, begin by simplifying the first two.
Notice we get the same result by adding the three exponents in one step.
TRY IT #1 Write each of the following products with a single base. Do not simplify further.
ⓐ ⓑ ⓒ
Using the Quotient Rule of Exponents
The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but
different exponents. In a similar way to the product rule, we can simplify an expression such as where
Consider the example Perform the division by canceling common factors.
Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.
In other words, when dividing exponential expressions with the same base, we write the result with the common base
and subtract the exponents.
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=34&chunk=81
|
In other words, when dividing exponential expressions with the same base, we write the result with the common base
and subtract the exponents.
For the time being, we must be aware of the condition Otherwise, the difference could be zero or negative.
Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify
matters and assume from here on that all variables represent nonzero real numbers.
The Quotient Rule of Exponents
For any real number and natural numbers and such that the quotient rule of exponents states that
EXAMPLE 2
Using the Quotient Rule
Write each of the following products with a single base. Do not simplify further.
1.2 • Exponents and Scientific Notation 25
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=34&chunk=82
|
ⓐ ⓑ ⓒ
Solution
Use the quotient rule to simplify each expression.
ⓐ ⓑ ⓒ
TRY IT #2 Write each of the following products with a single base. Do not simplify further.
ⓐ ⓑ ⓒ
Using the Power Rule of Exponents
Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the
power rule of exponents. Consider the expression The expression inside the parentheses is multiplied twice
because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent
of 3.
The exponent of the answer is the product of the exponents: In other words, when raising an
exponential expression to a power, we write the result with the common base and the product of the exponents.
Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different
terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=35&chunk=83
|
terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a
term in exponential notation is raised to a power. In this case, you multiply the exponents.
The Power Rule of Exponents
For any real number and positive integers and the power rule of exponents states that
EXAMPLE 3
Using the Power Rule
Write each of the following products with a single base. Do not simplify further.
ⓐ ⓑ ⓒ
26 1 • Prerequisites
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=35&chunk=84
|
Solution
Use the power rule to simplify each expression.
ⓐ ⓑ ⓒ
TRY IT #3 Write each of the following products with a single base. Do not simplify further.
ⓐ ⓑ ⓒ
Using the Zero Exponent Rule of Exponents
Return to the quotient rule. We made the condition that so that the difference would never be zero or
negative. What would happen if In this case, we would use the zero exponent rule of exponents to simplify the
expression to 1. To see how this is done, let us begin with an example.
If we were to simplify the original expression using the quotient rule, we would have
If we equate the two answers, the result is This is true for any nonzero real number, or any variable representing
a real number.
The sole exception is the expression This appears later in more advanced courses, but for now, we will consider the
value to be undefined.
The Zero Exponent Rule of Exponents
For any nonzero real number the zero exponent rule of exponents states that
EXAMPLE 4
Using the Zero Exponent Rule
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=36&chunk=85
|
value to be undefined.
The Zero Exponent Rule of Exponents
For any nonzero real number the zero exponent rule of exponents states that
EXAMPLE 4
Using the Zero Exponent Rule
Simplify each expression using the zero exponent rule of exponents.
ⓐ ⓑ ⓒ ⓓ
Solution
Use the zero exponent and other rules to simplify each expression.
ⓐ
ⓑ
1.2 • Exponents and Scientific Notation 27
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=36&chunk=86
|
ⓒ
ⓓ
TRY IT #4 Simplify each expression using the zero exponent rule of exponents.
ⓐ ⓑ ⓒ ⓓ
Using the Negative Rule of Exponents
Another useful result occurs if we relax the condition that in the quotient rule even further. For example, can we
simplify When —that is, where the difference is negative—we can use the negative rule of exponents to
simplify the expression to its reciprocal.
Divide one exponential expression by another with a larger exponent. Use our example,
If we were to simplify the original expression using the quotient rule, we would have
Putting the answers together, we have This is true for any nonzero real number, or any variable representing
a nonzero real number.
28 1 • Prerequisites
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=37&chunk=87
|
A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction
bar—from numerator to denominator or vice versa.
We have shown that the exponential expression is defined when is a natural number, 0, or the negative of a natural
number. That means that is defined for any integer Also, the product and quotient rules and all of the rules we will
look at soon hold for any integer
The Negative Rule of Exponents
For any nonzero real number and natural number the negative rule of exponents states that
EXAMPLE 5
Using the Negative Exponent Rule
Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.
ⓐ ⓑ ⓒ
Solution
ⓐ ⓑ
ⓒ
TRY IT #5 Write each of the following quotients with a single base. Do not simplify further. Write answers
with positive exponents.
ⓐ ⓑ ⓒ
EXAMPLE 6
Using the Product and Quotient Rules
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=38&chunk=88
|
Solution
ⓐ ⓑ
ⓒ
TRY IT #5 Write each of the following quotients with a single base. Do not simplify further. Write answers
with positive exponents.
ⓐ ⓑ ⓒ
EXAMPLE 6
Using the Product and Quotient Rules
Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.
ⓐ ⓑ ⓒ
Solution
ⓐ ⓑ
ⓒ
TRY IT #6 Write each of the following products with a single base. Do not simplify further. Write answers
with positive exponents.
ⓐ ⓑ
Finding the Power of a Product
To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents,
which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider
We begin by using the associative and commutative properties of multiplication to regroup the factors.
1.2 • Exponents and Scientific Notation 29
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=38&chunk=89
|
In other words,
The Power of a Product Rule of Exponents
For any real numbers and and any integer the power of a product rule of exponents states that
EXAMPLE 7
Using the Power of a Product Rule
Simplify each of the following products as much as possible using the power of a product rule. Write answers with
positive exponents.
ⓐ ⓑ ⓒ ⓓ ⓔ
Solution
Use the product and quotient rules and the new definitions to simplify each expression.
ⓐ ⓑ
ⓒ ⓓ
ⓔ
TRY IT #7 Simplify each of the following products as much as possible using the power of a product rule.
Write answers with positive exponents.
ⓐ ⓑ ⓒ ⓓ ⓔ
Finding the Power of a Quotient
To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the
power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following
example.
Let’s rewrite the original problem differently and look at the result.
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=39&chunk=90
|
power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following
example.
Let’s rewrite the original problem differently and look at the result.
It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.
30 1 • Prerequisites
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=39&chunk=91
|
The Power of a Quotient Rule of Exponents
For any real numbers and and any integer the power of a quotient rule of exponents states that
EXAMPLE 8
Using the Power of a Quotient Rule
Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with
positive exponents.
ⓐ ⓑ ⓒ ⓓ ⓔ
Solution
ⓐ ⓑ
ⓒ ⓓ
ⓔ
TRY IT #8 Simplify each of the following quotients as much as possible using the power of a quotient rule.
Write answers with positive exponents.
ⓐ ⓑ ⓒ ⓓ ⓔ
Simplifying Exponential Expressions
Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the
expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.
EXAMPLE 9
Simplifying Exponential Expressions
Simplify each expression and write the answer with positive exponents only.
ⓐ ⓑ ⓒ ⓓ
ⓔ ⓕ
1.2 • Exponents and Scientific Notation 31
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=40&chunk=92
|
Solution
ⓐ
ⓑ
ⓒ
ⓓ
ⓔ
ⓕ
TRY IT #9 Simplify each expression and write the answer with positive exponents only.
ⓐ ⓑ ⓒ ⓓ
ⓔ ⓕ
Using Scientific Notation
Recall at the beginning of the section that we found the number when describing bits of information in digital
images. Other extreme numbers include the width of a human hair, which is about 0.00005 m, and the radius of an
32 1 • Prerequisites
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=41&chunk=93
|
electron, which is about 0.00000000000047 m. How can we effectively work read, compare, and calculate with numbers
such as these?
A shorthand method of writing very small and very large numbers is called scientific notation, in which we express
numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the
first digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places n that you
moved the decimal point. Multiply the decimal number by 10 raised to a power of n. If you moved the decimal left as in a
very large number, is positive. If you moved the decimal right as in a small large number, is negative.
For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which
is 2.
We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=42&chunk=94
|
is 2.
We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive
because we moved the decimal point to the left. This is what we should expect for a large number.
Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the
same series of steps as above, except move the decimal point to the right.
Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent
of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for
a small number.
Scientific Notation
A number is written in scientific notation if it is written in the form where and is an integer.
EXAMPLE 10
Converting Standard Notation to Scientific Notation
Write each number in scientific notation.
ⓐ Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=42&chunk=95
|
EXAMPLE 10
Converting Standard Notation to Scientific Notation
Write each number in scientific notation.
ⓐ Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m
ⓑ Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m
ⓒ Number of stars in Andromeda Galaxy: 1,000,000,000,000
ⓓ Diameter of electron: 0.00000000000094 m
ⓔ Probability of being struck by lightning in any single year: 0.00000143
Solution
ⓐ
ⓑ
1.2 • Exponents and Scientific Notation 33
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=42&chunk=96
|
ⓒ
ⓓ
ⓔ
Analysis
Observe that, if the given number is greater than 1, as in examples a–c, the exponent of 10 is positive; and if the number
is less than 1, as in examples d–e, the exponent is negative.
TRY IT #10 Write each number in scientific notation.
ⓐ U.S. national debt per taxpayer (April 2014): $152,000
ⓑ World population (April 2014): 7,158,000,000
ⓒ World gross national income (April 2014): $85,500,000,000,000
ⓓ Time for light to travel 1 m: 0.00000000334 s
ⓔ Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715
Converting from Scientific to Standard Notation
To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal places
to the right if is positive or places to the left if is negative and add zeros as needed. Remember, if is positive, the
value of the number is greater than 1, and if is negative, the value of the number is less than one.
EXAMPLE 11
Converting Scientific Notation to Standard Notation
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=43&chunk=97
|
value of the number is greater than 1, and if is negative, the value of the number is less than one.
EXAMPLE 11
Converting Scientific Notation to Standard Notation
Convert each number in scientific notation to standard notation.
ⓐ ⓑ ⓒ ⓓSolution
ⓐ ⓑ ⓒ ⓓ
34 1 • Prerequisites
Access for free at openstax.org
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=43&chunk=98
|
TRY IT #11 Convert each number in scientific notation to standard notation.
ⓐ ⓑ ⓒ ⓓ
Using Scientific Notation in Applications
Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than
doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water.
Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around
molecules of water and 1 L of water holds about average drops. Therefore, there are
approximately atoms in 1 L of water. We simply multiply the decimal
terms and add the exponents. Imagine having to perform the calculation without using scientific notation!
When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For
example, consider the product The answer is not in proper scientific notation
|
Algebra-and-Trigonometry-2e-WEB.pdf#page=44&chunk=99
|
End of preview. Expand
in Data Studio
README.md exists but content is empty.
- Downloads last month
- 7