Mathematics 3104B Exponents and Logarithms

Adult Basic EducationMathematics

Mathematics 3104B

Exponents and LogarithmsCurriculum GuidePrerequisites: Math 1104A, Math 1104B and Math 1104C Math 2104A, Math 2104B and Math 2104C

Math 3104A

Credit Value: 1

Required Mathematics Courses [Degree and Technical Profile]Mathematics 1104AMathematics 1104BMathematics 1104CMathematics 2104AMathematics 2104BMathematics 2104CMathematics 3104AMathematics 3104BMathematics 3104C

Table of Contents

Introduction to Mathematics 3104B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPrerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTextbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTechnology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vCurriculum Guides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viStudy Guides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiResources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiRecommended Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Unit 1: Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Page 2

Unit 2: Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Page 4

Unit 3: Logarithms and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Page 6

Unit 4: Real Situations Involving Exponential and Logarithmic Functions . . . . . . . . . . . Page 10

Unit 5: Exponential Functions and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Page 14

Unit 6: Logarithms and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Page 16

Unit 7: Exponential and Logarithmic Equations and Identities . . . . . . . . . . . . . . . . . . . . . Page 18

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Page 23

Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Page 43

Curriculum Guide Mathematics 3104Biv

To the Instructor

I. Introduction to Mathematics 3104B

This course is an introduction to exponential and logarithmic functions, equations andgraphs which is an essential component of post-secondary Math courses.

The student will learn to interpret information from equations, graphs and writtendescriptions and to translate between these different ways of presentation. Other topicsinvolve a variety of strategies for simplifying or solving exponential and logarithmicequations. Real-life situations demonstrating exponential or logarithmic behavior includepopulation growth (human or bacteria), radioactive decay, earthquake and sound intensityand PH levels.

II. Prerequisites

Students taking this course need to be able to evaluate expressions involving integerexponents. They should be able to convert a percent into a decimal, and calculate percentincreases or decreases. There are exercises, (with solutions) for each Section of thetextbook in Prerequisites in the Teacher’s Resource Book. The instructor may use theseproblems to determine the student’s ability. In the Study Guide, students are oftendirected to see the instructor for Prerequisite exercises. The instructor, however, mustensure that the student has mastered these concepts before beginning a particular unit.

III. Textbook

Note: The Study Guide for this course has an Appendix A, which contains copies ofpages scanned from Mathematics 10.

Most of the concepts are introduced, developed and explained in the Examples. Theinstructor must insist that the student carefully studies and understands each Examplebefore moving on to the Exercises. In the Study Guide, the student is directed to see theinstructor if there are any difficulties.

There are four basic categories included in each section of the textbook which require thestudent to complete questions:

1. Investigate2. Discussing the Ideas3. Exercises4. Communicating the Ideas

Curriculum Guide Mathematics 3104Bv

To the Instructor

Investigate: This section looks at the thinking behind new concepts. The answers to itsquestions are found in the back of the text.

Discussing the Ideas: This section requires the student to write a response whichclarifies and demonstrates understanding of the concepts introduced. The answers tothese questions are not in the student text but are in the Teacher’s Resource Book. Therefore, in the Study Guide, the student is directed to see the instructor for correction.This will offer the instructor some perspective on the extent of the student’sunderstanding. If necessary, reinforcement or remedial work can be introduced. Studentsshould not be given the answer key for this section as the opportunity to assess thestudent’s understanding is then lost.

Exercises: This section helps the student reinforce understanding of the conceptsintroduced. There are three levels of Exercises:

A: direct application of concepts introduced B: multi-step problem solving and some real-life situationsC: problems of a more challenging nature

The answers to the Exercises questions are found in the back of the text.

Communicating the Ideas: This section helps confirm the student’s understanding of aparticular lesson by requiring a clearly written explanation. The answers toCommunicating the Ideas are not in the student text, but are in the Teacher’s ResourceBook. In the Study Guide students are asked to see the instructor for correction.

IV. Technology

It is important that students have a scientific calculator for their individual use. Ensurethat the calculators used have the word “scientific” on it as there are calculators designedfor calculation in other areas such as business or statistics which would not have thefunctions needed for study in this area.

A graphing calculator should be available to the students since the text provides manyopportunities for its use. The Teacher’s Resource Book suggests many opportunities touse a graphing calculator. These suggestions are outlined where there is the headingIntegrating Technology. In the Study Guide, students are directed to see the instructor

Curriculum Guide Mathematics 3104Bvi

To the Instructor

when a graphing calculator is required. The Teacher’s Resource Book contains a modulecalled Graphing Calculator Handbook which will help the instructor and student getacquainted with some of the main features of the TI-83 Plus graphing calculator.

Graphing software such as Graphmatica or Winplot can also be used if students do nothave access to a graphing calculator but do have access to a computer. The textbookdoes not offer the same guidance for graphing with these tools as it does for a graphingcalculator but each software program does have a HELP feature to answer questions.

V. Curriculum Guides

Each new ABE Mathematics course has a Curriculum Guide for the instructor and aStudy Guide for the student. The Curriculum Guide includes the specific curriculumoutcomes for the course. Suggestions for teaching, learning, and assessment are providedto support student achievement of the outcomes. Each course is divided into units. Eachunit comprises a two-page layout of four columns as illustrated in the figure below. Insome cases the four-column spread continues to the next two-page layout.

Curriculum Guide Organization:The Two-Page, Four-Column Spread

Unit Number - Unit Title Unit Number - Unit Title

Outcomes

Specificcurriculumoutcomes forthe unit.

Notes for Teaching andLearning

Suggested activities,elaboration of outcomes, andbackground information.

Suggestions for Assessment

Suggestions for assessingstudents’ achievement ofoutcomes.

Resources

Authorized andrecommendedresources thataddressoutcomes.

Curriculum Guide Mathematics 3104Bvii

To the Instructor

VI. Study Guides

The Study Guide provides the student with the name of the text(s) required for the courseand specifies the sections and pages that the student will need to refer to in order tocomplete the required work for the course. It guides the student through the course byassigning relevant reading and providing questions and/or assigning questions from thetext or some other resource. Sometimes it also provides important points for students tonote. (See the To the Student section of the Study Guide for a more detailed explanationof the use of the Study Guides.) The Study Guides are designed to give students somedegree of independence in their work. Instructors should note, however, that there ismuch material in the Curriculum Guides in the Notes for Teaching and Learning andSuggestions for Assessment columns that is not included in the Study Guide andinstructors will need to review this information and decide how to include it.

VII. Resources

Essential Resources

Addison Wesley Mathematics 12 (Western Canadian edition) Textbook ISBN:0-201-34624-9

Mathematics 12 Teacher’s Resource Book (Western Canadian edition) ISBN: 0-201-34626-5

Math 3104B Study Guide

Recommended Resources

Mathematics 12 Independent Study Guide (Western Canadian edition) ISBN: 0-201-34625-7

Center for Distance Learning and Innovation: http://www.cdli.caWinplot: http://math.exeter.edu/rparris/winplot.html

(Free graphing software)Graphmatica (Evaluation software available on CD-ROM contained in

Teacher’s Resource Book)CD Rom accompanying Teacher’s Resource Book

This CD contains selected solutions from the text and self test solutionsfrom the Independent Study Guide.

Curriculum Guide Mathematics 3104Bviii

To the Instructor

Other Resources Math Links: http://mathforum.org

http://spot.pcc.edu/~ssimonds/winplot (Free videos concerning Winplot)

http://www.pearsoned.ca/school/math/math/

http://www.ed.gov.nl.ca/edu/sp/mathlist.htm(Math Companion 3204 Supporting Document)

VIII. Recommended Evaluation

Written Notes 10%Assignments 10%Test(s) 30%Final Exam (entire course) 50%

100%

The overall pass mark for the course is 50%.

Exponents and Logarithms

Unit 1: Rational Exponents

Curriculum Guide Mathematics 3104BPage 2

Outcomes

1.1 Solve problems involvingrational exponents.

1.1.1 Simplify and evaluateexpressions containing integerexponents.

1.1.2 Apply the exponent lawsfor powers or variables with rationalexponents.

Notes for Teaching and Learning

This is a review of the exponent laws which are animportant prerequisite to the complete understanding ofrational exponents.

Given , the x under the root symbol is called thexn

radicand, and the small n indicating the root is called theindex.For example, given , 125 is the radicand and 3 is1253

the index.

Students should see that the exponent laws learned earlierfor integer exponents also apply when the exponent is arational number (fraction).

Students may need to be reminded that a power with arational exponent can be written in two different ways.For example :

8 8 82

31

3 2 3 2= =( ) ( )and

8 8 82

3 2 13 23= =( )

Unit 1: Rational Exponents



Study Guide questions 1.1 to 1.4 will meet the objectives ofOutcome 1.1.

See the Teacher’s Resource Book, Mathematics 10, pages 4 - 9,for extra problems that can be used for assessment or review.

See the Independent Study Guide, Mathematics 10, pages 29 - 36for Key Terms, Comprehension Questions, a Self-Test andFurther Practice that can be used for assessment or review.

There are three Worksheets (with answers) in Appendix B thatcould be used for review.

• Rational Exponents• Equations with Rational Exponents• Exponential Equations

Resources

Appendix A, pages 23 - 42

Appendix B, pages 43 - 50

Teacher’s Resource Book,Mathematics 10, Chapter 2, pages 4 - 9

Independent Study Guide,Mathematics 10, pages 29 -36

www.cdli.caMath 3204, Unit 03,Sections 03 and 04

Unit 2: Exponential Functions


Outcomes

2.1 Use exponential functions tosolve problems.

2.1.1 Give a definition of: (i) exponential function(ii) exponential growth(iii) exponential decay

2.1.2 Compare the equationsand graphs of exponential growthand exponential decay, and statethe similarities and differences.

2.1.3 Use technology to developa model to describe exponentialdata.

2.1.4 Analyze and solveproblems using the above models.


In this introduction to exponential functions and graphs,students should note the components of the function andthe characteristic shape of the exponential graph.

The textbook gives a variety of applications involvingexponential behavior, and asks the student to find specificinformation by interpreting it from the graph or bycalculating it through the function.

Unit 2: Exponential Functions





See the Independent Study Guide, Mathematics 12, pages 29 - 40for Key Terms, Comprehension Questions, a Self - Test andFurther Practice that can be used for assessment or review.

Resources

Mathematics 12 Section 2.1, Introduction toExponential Functions, pages 66 - 73

Teacher’s Resource Book,Mathematics 12, Chapter 2,pages 5 - 8


Unit 3: Logarithms and Their Properties


Outcomes

3.1 Describe the relationshipbetween a logarithmic andexponential function.

3.1.1 Define the term logarithm.( )loga x

3.1.2 State the relationshipbetween an exponential and alogarithm function.

3.1.3 Evaluate logarithms.

3.1.4 Convert from exponentialform to logarithmic form and viceversa.

Notes for Teaching And Learning

Hint: In Visualizing, on page 76 in Mathematics 12, thebase and the exponent are identified in both the log andexponential equation. To help students remember theirplacement - in the exponential form at least - theinstructor might suggest that the base could be referred toas “bacon” and the exponent as “eggs” to support the ideathat “ ” go together.baconeggs






See the Independent Study Guide, Mathematics 12, pages 29 - 40for Key Terms, Comprehension Questions, a Self - Test andFurther Practice that can be used for assessment or review.

Resources

Section 2.2, Defining aLogarithm, pages 74 - 78





Outcomes

3.2 Use the laws of logarithms tosimplify expressions and solveexponential equations.

3.2.1 State the law of logarithmsfor powers.

log logan

ax n x=

3.2.2 Use the law of logarithmsfor powers to remove a variablefrom the position of the exponent.

3.2.3 Solve exponentialequations by taking the log of bothsides of the equation.

3.2.4 State the law of logarithmsfor multiplication. log ( ) log loga a axy x y= +

3.2.5 State the law of logarithmsfor division.

log ( ) log loga a a

xy

x y= −

3.2.6 Use the laws of logarithmsto simplify expressions.

3.2.7 Evaluate log expressions.


It is not necessary that students be able to do thederivation for the log laws. It would be a worthwhileactivity for the instructor to go through the derivationwith the student for the purpose of demonstrating how alaw was developed and showing some techniques ofproblem solving.

The law of logarithms for powers will provide a means ofsolving for the exponent when it is the unknown or thedesired variable.

Instructors should reinforce the idea that an equation is abalance between the left and the right sides of the equalsign. Anything can be done to the equation as long as it isdone to both sides. The log is introduced on one side as ameans of accessing the law of logarithms for powers but itmust also be introduced on the other side in order tomaintain this balance.





See the Teacher’s Resource Book, Mathematics 12, pages 12 to14, for extra problems that can be used for assessment or review.

See the Independent Study Guide, Mathematics 12, pages 29 to40 for Key Terms, Comprehension Questions, a Self - Test andFurther Practice that can be used for assessment or review.

Resources

Mathematics 12, Section 2.3, The Laws ofLogarithms, pages 79 - 85



www.cdli.caMath 3204, Unit 03,Section 06

Unit 4: Real Situations Involving Exponential and Logarithmic Functions


Outcomes

4.1 Use exponential functions tosolve problems involving half-livesand doubling times.

4.1.1 State the role of A, b and xin the exponential equations

when b > 1 and when y Ab x=0 < b < 1.

4.1.2 Determine exponentialgrowth and decay equations forspecific situations.

4.1.3 Use log laws to solvegrowth and decay equations.

4.1.4 Use exponential functionsto find specific information ingrowth and decay problems.


When something is growing exponentially, it willbecome twice as large within a fixed length of time. Mostof the exponential growth problems involve functions ofthe type, y = Abx where A = initial amount , x = time, the base. b =(b > 1 for exponential growth) The doubling time depends on the base, b, and not on A.

The doubling equation has the form wherey Atd= ( )2

A = initial amount , t = elapsed time, d = doubling time

Either or the above forms can be used depending on theinformation given in the problem.

A similar property applies when something is decayingexponentially. It will become one-half as large within afixed period of time. The equations used can take theform of y = Abx , ( 0 < b < 1 for exponential decay).

The half-life equation has the form where y Ath= ( . )0 5

A = initial amountt = time elapsedh = half-lifeInstructors should ensure that students understand that theamount of time required for a population to double in agrowth situation remains constant. It does not depend onthe initial amount but only on the growth factor. Similarly, in a decay situation, the time required for apopulation to become half its original size is constant.Students may need to be reminded that when solvingexponential equations, such as the one in Example 1 onpage 87, the constant (2.28 in this example) should not bemultiplied by the base (1.014). The order of operationsspecifies that the exponent must be evaluated beforemultiplying.







Resources

Mathematics 12, Section 2.4 , Modeling RealSituations UsingExponential Functions: Part 1, pages 86-94





Outcomes

4.2 Use exponential equationsand logarithms to model realsituations.

4.2.1 Define the following term:i) logarithmic scale.

4.2.2 Create exponentialequations from word problems thatgive varied information.

4.2.3 Using exponentialinformation, find comparativeanswers.


Sometimes specific information, such as the initial size ofa population, is not known; yet it is still possible to find ananswer that is comparative. The answer describes anincrease or decrease in population size over a period oftime.

This comparative method is widely used when describingearthquake intensity, sound intensity and alkalinity ofsolutions.

Note: This section has appropriate applications for othercourses: knowledge of the pH scale will be needed forAcids and Bases in Chemistry and the decibel scale willbe needed for Sound in Physics.

All exponential equations in this section will have thesame type of information though the specific details willvary depending on the problem. In the case of population, there will be a need for twopopulation references in the formula; the initialpopulation, P0 , and the population at a later time, P. Theconstant increase will be the base. There will be twodifferent times; one will be the time for the constantincrease and the other will be the elapsed time. The instructors should work through several exampleswith the students and point out the significant informationwhich is the first critical step of problem solving.




Study Guide questions 4.5 and 4.6 will meet the objectives ofOutcome 4.2.



Resources

Mathematics 12 ,Section 2.5, Modeling RealSituations UsingExponential Functions: Part II, pages 95 - 101



Unit 5: Exponential Functions and Their Graphs


Outcomes

5.1 Graph and analyzeexponential functions, with andwithout technology.

5.1.1 Define the followingterms:

i) asymptoteii) vertical interceptiii) horizontal intercept

5.1.2 Identify the followingproperties from the graph of anexponential function of theform :f x b x( ) =

i) increasing or decreasing ii) vertical interceptiii) horizontal interceptiv) asymptotev) domain and range


Instructors should encourage students to look for thepattern in the table of values given for the function,

on page 104 in Mathematics 12.f x x( ) = 2

Translating graphs of functions was taught in previouscourses. Students should recognize transformations byinspecting the equation of the function. The three mostcommon transformations with exponential functions are:reflections, vertical stretches and horizontal stretches.

If students have difficulty, instructors should assignPrerequisite exercises on page 22 in the Teacher’sResource Book. If necessary, Chapter 1 in Mathematics12 should be reviewed by the student.

The Law of Exponents for multiplication is verified onthe top of page 105 in Mathematics 12.

Note: The asymptote for equation of the form is described as the x-axis. The equation of thef x b x( ) =

x-axis is y = 0.

Unit 5: Exponential Functions and Their Graphs






Resources

Mathematics 12, Section 2.6, Analyzing theGraphs of ExponentialFunctions, pages 104 - 112



Unit 6: Logarithms and Their Graphs


Outcomes

6.1 Graph and analyze alogarithmic function, with andwithout technology.

6.1.1 Show that taking theinverse of an exponential functioncreates a logarithmic function.

6.1.2 Graph an exponentialfunction and its inverse byinterchanging points and byreflecting the graph across the liney = x.

6.1.3 Graph logarithmicfunctions using a table of values,technology and transformations.

6.1.4 Identify the followingproperties from the graph of anlogarithmic function of the form

f x xb( ) log=(b > 0, b … 1) .

i) increasing or decreasing ii) vertical interceptiii) horizontal interceptiv) asymptotev) domain and range

6.1.5 Compare the properties ofthe graphs of exponential functionsand logarithmic functions.


The instructor should assign Prerequisite exercises onpage 36 in the Teacher’s Resource Book, Mathematics 12,Chapter 2, if students need review on the properties ofinverse functions and how to find the inverse of afunction algebraically and graphically.

If the word “inverse” is thought of as reflection, thenwhen looking at the tables of values for an exponentialfunction and its inverse, (a logarithmic function), studentsshould see that the x and y values are interchanged.

When the graphs of an exponential function and alogarithmic function are compared, it will be clear thatthey are reflections of each other across the line y = x.

Unit 6: Logarithms and Their Graphs






Resources

Mathematics 12, Section 2.10, LogarithmicFunctions, pages 133 - 140



Unit 7: Exponential and Logarithmic Equations and Identities


Outcomes

7.1 Find exact roots forexponential equations.

7.1.1 Describe the differencebetween approximate numbers andexact numbers.

7.1.2 Solve exponentialequations by expressing (whenpossible) both sides with the samebase.

7.1.3 Solve exponentialequations by taking the logarithmof both sides.


When finding exact answers, no technology will be used.In other words, calculators will not be needed. Instructorsshould ensure that students understand the format of theanswer required in this section.

Exercise 6 of the Investigate on page 146 ofMathematics 12 will show that not all equations can besolved using all 5 approaches.

The Investigate, Discussing the Ideas andCommunicating the Ideas are very useful sections forthis unit. A student must be aware of different strategies,and be able to assess an equation to determine which strategy is appropriate or preferable.

Instructors should encourage the students to spend timeon these activities and ensure that Investigate andDiscussing the Ideas are corrected before the studentsmove on to Exercises. By the time they reachCommunicating the Ideas, students should be able toanswer the questions with knowledge and experience.







Resources

Mathematics 12Section 2.11, RevisitingExponential Functions,pages 146 - 149

Teacher’s Resource Book,Mathematics 12, Chapter 2, pages 42 - 44




Outcomes

7.2 Use the definition of alogarithm and the laws oflogarithms to solve logarithmicequations and identities.

7.2.1 Define the following:i) identityii) extraneous root

7.2.2 Solve equations whichhave both sides expressed aslogarithms with the same base.

7.2.3 Solve equations whichhave some terms expressed aslogarithms and some as numbers.

7.2.4 Verify that the solutions tologarithmic equations are valid.

7.2.5 Prove identities using lawsof logarithms.


When solving log equations, students should be guided to check the solutions in the original equation to ensure thateach solution is valid.

In Example 1a) on page 150 in the Mathematics 12, x = !2 is an extraneous root because when this value issubstituted into the first term it becomes , whichlog ( )5 1−is undefined. Instructors should ensure that students donot generalize from this example to conclude that allnegative values are extraneous.

Supplementary Examples, 1b), on page 45 in Teacher’sResource Book is an example of an equation which has anegative value for an answer.







Questions chosen from Masters 2.4 to 2.8 could be used on a unittest.

Resources

Mathematics 12, Section 2.12, LogarithmicEquations and Identities,pages 150 -153

Teacher’s Resource Book,Mathematics 12, Chapter 2,pages 44 - 47Masters 2.4 - 2.8


Appendix AExponents

©Reproduced with the permission of Pearson Education Canada.








Exercises





Visualizing

Volume 8000 cm3

20 cmArea

3600 cm2 60 cm





Area30 cm2





Exercises

a) b) c)

Area9.61 m2

Area8.5 m2

Area6.4 m2



6. Use the volume of each cube. Determine the length of an edge and the area of a face. Givethe answer to 1 decimal place, where necessary.

a) b) c)

Volume1000 cm3

Volume250 cm3

Volume125 cm3







18

Visualizing

Volume Volume 8 cm3 2 cm cm3 ½ cm

8 8 213 3= = 8 1

812

13

3

−= =



Visualizing



Exercises





13. a) A cube has a volume of 8000 cm3. Determine the length of each edge and the area of each face.

b) Repeat part a for a cube with a volume of 2 m3. Express your answers in radical form.

c) Repeat part a for a cube with a volume of V cubic metres.

14. a) The area of each face of a cube is 25 cm2. Determine the length of each edge and the volume of the cube.

» 25 cm2 » 2 m2 » A m2

Volume8000 cm3 V m3Volume

2 m3

Appendix BWorksheets


Rational Exponents

Write in radical form.

1. 2. 3. 4. 5. 6. 213 37

32 x

12 a

15 6

43 6

34

7. 8. 9. 10. 11. 12. 712

−9

15

−x

− 37 b

− 65 ( )3

12x 3

12x

Write using exponents.

13. 14. 15. 16. 17. 18. 7 34 − 113 a 25 643 ( )b3 4

19. 20. 21. 22. 23. 24. 1x

13 a

15 4( )x

2 33 b 3 5x 5 34 t

Evaluate without using a calculator..

25. 26. 27. 28. 29. 30. 412 125

13 16

14

−( )−32

15 250 5. ( )−

−27

13

31. 32. 33. 34. 35. 36. ( )6416

−0 04

12. 810 25. 0 001

13. 4

9

12⎛

⎝⎜⎞⎠⎟

−−

⎛⎝⎜

⎞⎠⎟

278

13

37. 38. 39. 40. 41. 42. 823 4

32 92 5. 81

34 16

34

−( )−32

25

43. 44. 45. 46. 47. 48. ( )−−

853 ( )−

−27

23 1

53 ( )−

−1

85 100

9

32⎛

⎝⎜⎞⎠⎟

278

23⎛

⎝⎜⎞⎠⎟

−


Simplify without using a calculator.

49. a) b) 50. a) b) 912 9

12

−8

13 8

13

−

51. a) b) 52. a) b) 823 8

23

−16

14 16

14

−

53. 54. 55. 56. 2713

−27

23 4

32 25

12

−

57. 58. 59. 60. 8112 49

12

−4

32 16

34

61. 62. 63. 64. ( )−−

12513 4 0 5− . − 8

23 ( )5

13 3−

65. 66. ( )16 5120− ( )9 16

12

12 2+


Equations with Rational Exponents

Give the power to which you would raise both sides of each equation in order tosolve the equation.

1. 2. 3. 4. x12 9= x

23 4= x

−=

13 2 x

− −=34 18

Solve each equation.

5. a) b) a34 8= ( )3 1 8

34x + =

6. a) b) y−

=12 6 ( )3 6

12y

−=

7. a) b) 2 1012y

−= ( )2 10

12y

−=

8. a) b) ( )9 423t

−= 9 4

23t

−=

9. 10. ( )8 413− =y ( )3 1 1

4

23n − =

−

11. 12. ( )x 2234 25+ = ( )x 2

129 5+ =


Exponential Equations

Solve each of the following equations.

1. 2. 3. 4. 3 19

x = 25 125x = 4 18

x = 36 6x =

5. 6. 7. 8. 3 127

x = 5 125x = 8 22+ =x 4 81− =x

9. 10. 11. 12. 27 32 1x− = 49 7 72x− = 4 162 5 1x x+ += 3 95 4− + =( )x x

13. 14. 15. 25 52 6x x= + 6 361 1x x+ −= 10 1001 4x x− −=


Solutions

Rational Exponents

1. 2. 3. 4. 5. 6. 23 ( )37 3 x a5 ( )63 4 ( )63 3

7. 8. 9. 10. 11. 12. 17

195

137 x

165 b

3x 3 x

13. 14. 15. 16. 17. 18. 712 34

12 ( )−11

13 a

25 6

43 b

43

19. 20. 21. 22. 23. 24. x− 1

2 a− 1

3 x− 4

5 213 b 3

12

52x 5

14

34t

25. 26. 27. 28. 29. 30. 2 5 12

− 2 5 − 13

31. 32. 33. 34. 35. 36. 12

0 2. 3 01. 23

32

37. 38. 39. 40. 41. 42. 4 8 243 27 18

4

43. 44. 45. 46. 47. 48. − 132

19

1 1 100027

49


49. a) 3 b) 50. a) 2 b) 51. a) 4 b) 13

12

14

52. a) 2 b) 53. 54. 9 55. 8 56. 12

13

15

57. 9 58. 59. 8 60. 8 61. 62. 17

− 15

12

63. !4 64. 65. 66. 4915

12


Equations with Rational Exponents

1. 2 2. 3. 4. 32

− 3 − 43

5. a) 16 b) 5 6. a) b) 136

1108

7. a) b) 8. a) b) 125

1200

172

278

9. !56 10. 3 11. ±11 12. ±4

Exponential Equations

1. 1 2. 3. 4. 5. !3 6. 32

12

14

32

7. 8. 9. 10. 11. No solution− 53

− 12

23

114

12. 13. 2 14. 3 15. 3− 59

Documents

Mathematics 3104B Exponents and Logarithms