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Algebra and Trigonometry 2e
SENIOR CONTRIBUTING AUTHOR
JAY ABRAMSON, ARIZONA STATE UNIVERSITY | Algebra-and-Trigonometry-2e-WEB.pdf#page=2&chunk=0 |
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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
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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... | 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 37... | 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
Expon... | 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
... | 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 95... | 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 o... | Algebra-and-Trigonometry-2e-WEB.pdf#page=8&chunk=13 |
Exercises 1313
Proofs, Identities, and Toolkit Functions 1321A
Answer Key 1333
Index 1503
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Preface
About OpenStax
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our mission to transform learning so that education works for every student. Through our partnerships with
philanthropic organizations and our alliance with other educati... | Algebra-and-Trigonometry-2e-WEB.pdf#page=10&chunk=15 |
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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 an... | 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 m... | 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: F... | 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 ... | 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 b... | 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 shiftin... | 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... | 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
Narr... | 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 th... | 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... | 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 Exerc... | 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 ... | 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 compilat... | 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 coordina... | 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
accou... | 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
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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 book... | 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 Faldu... | 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 Commun... | 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... | 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 Colleg... | 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 Antar... | 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 Afr... | 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 iden... | 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... | 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 ... | 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 ... | 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) contai... | 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 opens... | 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 ... | 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 e... | 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 t... | 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 t... | 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 ... | 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 ... | 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:
... | 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 nu... | 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, rad... | 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 sim... | 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 ... | 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... | 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 Prope... | 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 m... | 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 te... | 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 prope... | 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 numb... | 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 ex... | 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 t... | 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 Valu... | 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 tr... | 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 Figur... | 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... | 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 num... | 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 ... | 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 join... | 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 eq... | 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 who... | 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 e... | 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” port... | 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 rul... | 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 ... | 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 qualify... | 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
p... | 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 t... | 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 ... | 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 expres... | 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 negati... | 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... | 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.... | 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 ex... | 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 • Prere... | 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.
ⓐ ... | 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,... | 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 notati... | 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, 0.00000... | 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 elec... | 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 po... | 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 calcu... | Algebra-and-Trigonometry-2e-WEB.pdf#page=44&chunk=99 |
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