Section 3.2: Negative Exponents


 Alexina Stephens
 4 years ago
 Views:
Transcription
1 Section 3.2: Negtive Exponents Objective: Simplify expressions with negtive exponents using the properties of exponents. There re few specil exponent properties tht del with exponents tht re not positive. The rst is considered in the following exmple, which is written out in two dierent wys. Exmple. Simplify. Use quotient rule; subtrct exponents 0 ; now consider the problem in the second wy Rewrite exponents; use repeted multipliction Reduce out ll the 0 s ; combine the two solutions; get: 0 Our Finl Solution This nl result is n importnt property known s the zero power rule of exponents. Zero Power Rule of Exponents: 0 Any nonzero number or expression rised to the zero power will lwys be. This is illustrted in the following exmple. Exmple 2. Simplify. (3x 2 ) 0 Zero power rule Here we re ssuming tht x is not 0. If x0, then (3 0 2 ) 0 would equl 0, not. Another property we will consider here dels with negtive exponents. Agin we will solve the following exmple in two wys. 85
2 Exmple 3. Simplify nd write the nswer using only positive exponents. 5 2 Use the quotient rule; subtrct exponents ; solve this problem nother wy Rewrite exponents s repeted multipliction Reduce three 0 s out of top nd bottom Simplify to exponents ; combine the two solutions; get: Our Finl Solution This exmple illustrtes n importnt property of exponents. Negtive exponents yield the reciprocl of the bse. Once we tke the reciprocl, the exponent is now positive. Also, it is importnt to note tht negtive exponent does not men tht the expression is negtive; only tht we need the reciprocl of the bse. The rules of negtive exponents follow. Rules of Negtive Exponents: m b b m Negtive exponents cn be combined in severl dierent wys. As generl rule, if we think of our expression s frction, negtive exponents in the numertor must be moved to the denomintor; likewise, negtive exponents in the denomintor need to be moved to the numertor. When the bse with exponent moves, the exponent is now positive. This is illustrted in the following exmple. Exmple 4. Simplify nd write the nswer using only positive exponents. b 2 c Move negtive exponents on b; d; nd e; 2d e 4 f 2 exponents become positive cde 4 2b 2 f 2 86
3 As we simplied our frction, we took specil cre to move the bses tht hd negtive exponent, but the expression itself did not become negtive becuse of those exponents. Also, it is importnt to remember tht exponents only eect the bse they re ttched to. The 2 in the denomintor of the bove exmple does not hve n exponent on it, so it does not move with the d. We now hve the following nine properties of exponents. It is importnt tht we re very fmilir with ll of them. Note tht when simplifying expressions involving exponents, the nl form is usully written using only positive exponents. Properties of Exponents n +n (b) m b m n m n b m b m m ( ) n n 0 b b m World View Note: Nicols Chuquet, the French mthemticin of the 5th century, wrote 2 m to indicte 2x. This ws the rst known use of the negtive exponent. Simplifying with negtive exponents is much the sme s simplifying with positive exponents. It is the dvice of the uthor to keep the negtive exponents until the end of the problem nd then move them round to their correct loction (numertor or denomintor). As we do this, it is importnt to be very creful of rules for dding, subtrcting, nd multiplying with negtives. This is illustrted in the following exmples. Exmple 5. Simplify nd write the nswer using only positive exponents. 4x 5 y 3 3x 3 y 2 6x 5 y 3 2x 2 y 5 6x 5 y 3 2x 3 y 8 2x 3 y 8 Simplify numertor with product rule; dd exponents Use quotient rule; subtrct exponents; be creful with negtives ( 2) ( 5) ( 2) ( 5) 3 ( 5) + ( 3) 8 Move negtive exponent to denomintor; exponent becomes positive 87
4 Exmple 6. Simplify nd write the nswer using only positive exponents. (3b 3 ) 2 b b b 6 b In numertor; use power rule with 2; multiply exponents In denomintor; b 0 In numertor; use product rule; dd exponents 3 2 b b 9 2 Use quotient rule; subtrct exponents; be creful with negtives ( ) ( 4) ( ) Move 3 nd b to denomintor becuse of negtive exponents; exponents become positive 3 2 2b 9 Evlute b 9 In the previous exmple it is importnt to point out tht when we simplied 3 2, we moved the three to the denomintor nd the exponent becme positive. We did not mke the number negtive! Negtive exponents never mke the bses negtive; they simply men we hve to tke the reciprocl of the bse. One finl exmple with negtive exponents is given here. Exmple 7. Simplify nd write the nswer using only positive exponents. 3x 2 y 5 z 3 6x 6 y 2 z 3 9(x 2 y 2 ) 3 3 In numertor; use product rule; dding exponents In denomintor; use power rule; multiplying exponents 8x 8 y 3 z 0 9x 6 y 6 3 Use quotient rule to subtrct exponents; be creful with negtives: ( 8) ( 6) ( 8) ( 6) 3 (2x 2 y 3 z 0 ) 3 Prentheses re done; use power rule with x 6 y 9 z 0 Move 2 with negtive exponent down nd z 0 ; exponent of 2 becomes positive x 6 y 9 Evlute x 6 y
5 3.2 Prctice Simplify ech expression. Your nswer should contin only positive exponents. ) 2x 4 y 2 (2xy 3 ) 4 2) 2 2 b 3 (2 0 b 4 ) 4 3) ( 4 b 3 ) 3 2 b 2 4) 2x 3 y 2 (2x 3 ) 0 5) (2x 2 y 2 ) 4 x 4 6) (m 0 n 3 2m 3 n 3 ) 0 7) (x 3 y 4 ) 3 x 4 y 4 8) 2m n 3 (2m n 3 ) 4 9) 0) 2x 3 y 2 3x 3 y 3 3x 0 3y 3 3yx 3 2x 4 y 3 ) 4xy 3 x 4 y 0 4y 2) 3) 3x 3 y 2 4y 2 3x 2 y 4 u 2 v 2u 0 v 4 2uv 4) 2xy2 4x 3 y 4 4x 4 y 4 4x 5) u 2 4u 0 v 3 3v 2 6) 2x 2 y 2 4yx 2 7) 2y (x 0 y 2 ) 4 8) (4 ) 4 2b 9) ( 22 b 3 ) 4 20) ( 2y 4 x 2 ) 2 2) 22) 2nm 4 (2m 2 n 2 ) 4 2y 2 (x 4 y 0 ) 4 23) (2mn)4 m 0 n 2 24) 2x 3 (x 4 y 3 ) 25) y3 x 3 y 2 (x 4 y 2 ) 3 26) 2x 2 y 0 2xy 4 (xy 0 ) 27) 2u 2 v 3 (2uv 4 ) 2u 4 v 0 28) 2yx2 x 2 (2x 0 y 4 ) 29) ( 2x0 y 4 y 4 ) 3 30) u 3 v 4 2v(2u 3 v 4 ) 0 3) y(2x4 y 2 ) 2 2x 4 y 0 32) 33) b (2 4 b 0 ) b 2 2yzx 2 2x 4 y 4 z 2 (zy 2 ) 4 34) 2b4 c 2 (2b 3 c 2 ) 4 2 b 4 35) 2kh0 2h 3 k 0 (2kj 3 ) 2 36) ( (2x 3 y 0 z ) 3 x 3 y 2 2x 3 ) 2 37) (cb3 ) b 2 ( b 2 c 3 ) 3 89
6 38) 2q4 m 2 p 2 q 4 (2m 4 p 2 ) 3 39) (yx 4 z 2 ) z 3 x 2 y 3 z 40) 2mpn 3 (m 0 n 4 p 2 ) 3 2n 2 p 0 90
7 ) 32x 8 y 0 2) 32b3 2 3) 25 b 4) 2x 3 y 2 5) 6x 4 y 8 6) 7) y 6 x ) m 5 n 5 9) 2 9y 0) y5 2x 7 ) y 2 x 3 2) y8 x 5 4 u 3) 4v 6 4) x7 y 2 2 u 2 5) 2v 5 6) y 2x 4 7) 2 y 7 8) 6 2b 9) 6 2 b 2 20) y8 x 4 4 2) 8m 4 n 7 22) 2x 6 y 2 23) 6n 6 m 4 24) 2x y 3 25) 3.2 Answers x 5 y 26) 4y 4 27) u 2v 28) 4y 5 29) 8 30) 2u 3 v 5 3) 2y 5 x 4 32) 3 2b 3 33) 34) 35) x 2 y z 2 8c 0 b 2 h 3 k j 6 36) x30 z 6 6y 4 37) 2b4 2 c 7 38) m4 q 8 4p 4 39) x 2 y 4 z 4 40) mn7 p 5 9
Section 3.1: Exponent Properties
Section.1: Exponent Properties Ojective: Simplify expressions using the properties of exponents. Prolems with exponents cn often e simplied using few sic exponent properties. Exponents represent repeted
More information5.2 Exponent Properties Involving Quotients
5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use
More informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
More informationExponentials  Grade 10 [CAPS] *
OpenStxCNX module: m859 Exponentils  Grde 0 [CAPS] * Free High School Science Texts Project Bsed on Exponentils by Rory Adms Free High School Science Texts Project Mrk Horner Hether Willims This work
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationLesson 25: Adding and Subtracting Rational Expressions
Lesson 2: Adding nd Subtrcting Rtionl Expressions Student Outcomes Students perform ddition nd subtrction of rtionl expressions. Lesson Notes This lesson reviews ddition nd subtrction of frctions using
More informationConsolidation Worksheet
Cmbridge Essentils Mthemtics Core 8 NConsolidtion Worksheet N Consolidtion Worksheet Work these out. 8 b 7 + 0 c 6 + 7 5 Use the number line to help. 2 Remember + 2 2 +2 2 2 + 2 Adding negtive number is
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationChapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
More informationMATHEMATICS AND STATISTICS 1.2
MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 200910 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More informationCH 9 INTRO TO EQUATIONS
CH 9 INTRO TO EQUATIONS INTRODUCTION I m thinking of number. If I dd 10 to the number, the result is 5. Wht number ws I thinking of? R emember this question from Chpter 1? Now we re redy to formlize the
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationQuotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m
Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors PreChpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More informationIdentify graphs of linear inequalities on a number line.
COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line.  When grphing firstdegree eqution, solve for the vrible. The grph of this solution will be single point
More informationTHE DISCRIMINANT & ITS APPLICATIONS
THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used
More informationSimplifying Algebra. Simplifying Algebra. Curriculum Ready.
Simplifying Alger Curriculum Redy www.mthletics.com This ooklet is ll out turning complex prolems into something simple. You will e le to do something like this! ( 9 # + 4 ' ) ' ( 9 + 7) ' ' Give this
More informationIntroduction to Group Theory
Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More information12.1 Introduction to Rational Expressions
. Introduction to Rtionl Epressions A rtionl epression is rtio of polynomils; tht is, frction tht hs polynomil s numertor nd/or denomintor. Smple rtionl epressions: 0 EVALUATING RATIONAL EXPRESSIONS To
More informationREVIEW Chapter 1 The Real Number System
Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole
More informationIntroduction to Algebra  Part 2
Alger Module A Introduction to Alger  Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger  Prt Sttement of Prerequisite
More informationLesson 2.4 Exercises, pages
Lesson. Exercises, pges A. Expnd nd simplify. ) + b) ( ) () 0  ( ) () 0 c) 7 + d) (7) ( ) 7  + 8 () ( 8). Expnd nd simplify. ) b)  7  + 7 7( ) ( ) ( ) 7( 7) 8 (7) P DO NOT COPY.. Multiplying nd Dividing
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More information4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve
Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions
More informationALevel Mathematics Transition Task (compulsory for all maths students and all further maths student)
ALevel Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length:  hours work (depending on prior knowledge) This trnsition tsk provides revision
More informationAdding and Subtracting Rational Expressions
6.4 Adding nd Subtrcting Rtionl Epressions Essentil Question How cn you determine the domin of the sum or difference of two rtionl epressions? You cn dd nd subtrct rtionl epressions in much the sme wy
More informationThe graphs of Rational Functions
Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior
More informationI do slope intercept form With my shades on MartinGay, Developmental Mathematics
AATA Dte: 1//1 SWBAT simplify rdicls. Do Now: ACT Prep HW Requests: Pg 49 #1745 odds Continue Vocb sheet In Clss: Complete Skills Prctice WS HW: Complete Worksheets For Wednesdy stmped pges Bring stmped
More informationSample pages. 9:04 Equations with grouping symbols
Equtions 9 Contents I know the nswer is here somewhere! 9:01 Inverse opertions 9:0 Solving equtions Fun spot 9:0 Why did the tooth get dressed up? 9:0 Equtions with pronumerls on both sides GeoGebr ctivity
More informationRead section 3.3, 3.4 Announcements:
Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f
More informationPART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.
PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic
More informationEquations and Inequalities
Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in
More informationPrecalculus Spring 2017
Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationVectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3dimensional vectors:
Vectors 1232018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2dimensionl vectors: (2, 3), ( )
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationAQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
More information7h1 Simplifying Rational Expressions. Goals:
h Simplifying Rtionl Epressions Gols Fctoring epressions (common fctor, & , no fctoring qudrtics) Stting restrictions Epnding rtionl epressions Simplifying (reducin rtionl epressions (Kürzen) Adding nd
More informationENGI 3424 Engineering Mathematics Five Tutorial Examples of Partial Fractions
ENGI 44 Engineering Mthemtics Five Tutoril Exmples o Prtil Frctions 1. Express x in prtil rctions: x 4 x 4 x 4 b x x x x Both denomintors re liner nonrepeted ctors. The coverup rule my be used: 4 4 4
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
More informationIn this skill we review equations that involve percents. review the meaning of proportion.
6 MODULE 5. PERCENTS 5b Solving Equtions Mening of Proportion In this skill we review equtions tht involve percents. review the mening of proportion. Our first tsk is to Proportions. A proportion is sttement
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationReview Factoring Polynomials:
Chpter 4 Mth 0 Review Fctoring Polynomils:. GCF e. A) 5 5 A) 4 + 9. Difference of Squres b = ( + b)( b) e. A) 9 6 B) C) 98y. Trinomils e. A) + 5 4 B) + C) + 5 + Solving Polynomils:. A) ( 5)( ) = 0 B) 4
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
More informationfractions Let s Learn to
5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin
More information378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
More informationpadic Egyptian Fractions
padic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Setup 3 4 pgreedy Algorithm 5 5 pegyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationLogarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.
Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the
More informationLoudoun Valley High School Calculus Summertime Fun Packet
Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!
More informationExample 1: Express as a sum of logarithms by using the Product Rule. (By the definition of log)
Section 5. Properties of Logrithmic Functions Section 5. Properties of Logrithmic Functions This section covers some properties of rithmic function tht re very similr to the rules for exponents. Properties
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities RentHep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationSect 10.2 Trigonometric Ratios
86 Sect 0. Trigonometric Rtios Objective : Understnding djcent, Hypotenuse, nd Opposite sides of n cute ngle in right tringle. In right tringle, the otenuse is lwys the longest side; it is the side opposite
More informationUniversitaireWiskundeCompetitie. Problem 2005/4A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that
Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de
More informationMA 131 Lecture Notes Calculus Sections 1.5 and 1.6 (and other material)
MA Lecture Notes Clculus Sections.5 nd.6 (nd other teril) Algebr o Functions Su, Dierence, Product, nd Quotient o Functions Let nd g be two unctions with overlpping doins. Then or ll x coon to both doins,
More informationIntroduction To Matrices MCV 4UI Assignment #1
Introduction To Mtrices MCV UI Assignment # INTRODUCTION: A mtrix plurl: mtrices) is rectngulr rry of numbers rrnged in rows nd columns Exmples: ) b) c) [ ] d) Ech number ppering in the rry is sid to be
More informationLesson Notes: Week 40Vectors
Lesson Notes: Week 40Vectors Vectors nd Sclrs vector is quntity tht hs size (mgnitude) nd direction. Exmples of vectors re displcement nd velocity. sclr is quntity tht hs size but no direction. Exmples
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationDo the onedimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent?
1 Problem 1 Do the onedimensionl kinetic energy nd momentum opertors commute? If not, wht opertor does their commuttor represent? KE ˆ h m d ˆP i h d 1.1 Solution This question requires clculting the
More informationMultiplying integers EXERCISE 2B INDIVIDUAL PATHWAYS. 6 ì 4 = 6 ì 0 = 4 ì 0 = 6 ì 3 = 5 ì 3 = 4 ì 3 = 4 ì 2 = 4 ì 1 = 5 ì 2 = 6 ì 2 = 6 ì 1 =
EXERCISE B INDIVIDUAL PATHWAYS Activity B Integer multipliction doc69 Activity B More integer multipliction doc698 Activity B Advnced integer multipliction doc699 Multiplying integers FLUENCY
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationPreSession Review. Part 1: Basic Algebra; Linear Functions and Graphs
PreSession Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationBases for Vector Spaces
Bses for Vector Spces 22625 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationMatrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24
Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationInfinite Geometric Series
Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to
More informationReversing the Chain Rule. As we have seen from the Second Fundamental Theorem ( 4.3), the easiest way to evaluate an integral b
Mth 32 Substitution Method Stewrt 4.5 Reversing the Chin Rule. As we hve seen from the Second Fundmentl Theorem ( 4.3), the esiest wy to evlute n integrl b f(x) dx is to find n ntiderivtive, the indefinite
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More informationx means to use x as a factor five times, or x x x x x (2 c ) means to use 2c as a factor four times, or
14 DAY 1 CHAPTER FIVE Wht fscinting mthemtics is now on our gend? We will review the pst four chpters little bit ech dy becuse mthemtics builds. Ech concept is foundtion for nother ide. We will hve grph
More informationPreparation for A Level Wadebridge School
Preprtion for A Level Mths @ Wdebridge School Bridging the gp between GCSE nd A Level Nme: CONTENTS Chpter Removing brckets pge Chpter Liner equtions Chpter Simultneous equtions 6 Chpter Fctorising 7 Chpter
More informationChapter 8.2: The Integral
Chpter 8.: The Integrl You cn think of Clculus s doulewide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in
More informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2 elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Instruction
Lesson 3: Creting Qudrtic Equtions in Two or More Vriles Prerequisite Skills This lesson requires the use of the following skill: solving equtions with degree of Introduction 1 The formul for finding the
More informationMATH FIELD DAY Contestants Insructions Team Essay. 1. Your team has forty minutes to answer this set of questions.
MATH FIELD DAY 2012 Contestnts Insructions Tem Essy 1. Your tem hs forty minutes to nswer this set of questions. 2. All nswers must be justified with complete explntions. Your nswers should be cler, grmmticlly
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationMAS 4156 Lecture Notes Differential Forms
MAS 4156 Lecture Notes Differentil Forms Definitions Differentil forms re objects tht re defined on mnifolds. For this clss, the only mnifold we will put forms on is R 3. The full definition is: Definition:
More information71: Zero and Negative Exponents
7: Zero nd Negtive Exponents Objective: To siplify expressions involving zero nd negtive exponents Wr Up:.. ( ).. 7.. Investigting Zero nd Negtive Exponents: Coplete the tble. Write nonintegers s frctions
More informationIntermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4
Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one
More informationIntroduction to Mathematical Reasoning, Saylor 111
Frction versus rtionl number. Wht s the difference? It s not n esy question. In fct, the difference is somewht like the difference between set of words on one hnd nd sentence on the other. A symbol is
More informationCHM Physical Chemistry I Chapter 1  Supplementary Material
CHM 3410  Physicl Chemistry I Chpter 1  Supplementry Mteril For review of some bsic concepts in mth, see Atkins "Mthemticl Bckground 1 (pp 596), nd "Mthemticl Bckground " (pp 109111). 1. Derivtion
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Algebr Opertions nd Epressions Common Mistkes Division of Algebric Epressions Eponentil Functions nd Logrithms Opertions nd their Inverses Mnipulting
More informationLecture 7 notes Nodal Analysis
Lecture 7 notes Nodl Anlysis Generl Network Anlysis In mny cses you hve multiple unknowns in circuit, sy the voltges cross multiple resistors. Network nlysis is systemtic wy to generte multiple equtions
More informationSection 5.5 from Basic Mathematics Review by Oka Kurniawan was developed by OpenStax College, licensed by Rice University, and is available on the
Section 5.5 from Bsic Mthemtics Review by Ok Kurniwn ws developed by OpenStx College, licensed by Rice University, nd vilble on the Connexions website. It used under Cretive Commons Attribution 3.0 Unported
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose
More informationElementary Mathematical Concepts and Operations
Elementry Mthemticl Concepts nd Opertions After studying this chpter you should be ble to: dd, subtrct, multiply nd divide positive nd negtive numbers understnd the concept of squre root expnd nd evlute
More informationMA Lesson 21 Notes
MA 000 Lesson 1 Notes ( 5) How would person solve n eqution with vrible in n eponent, such s 9? (We cnnot rewrite this eqution esil with the sme bse.) A nottion ws developed so tht equtions such s this
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationMath 426: Probability Final Exam Practice
Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by
More informationMATH STUDENT BOOK. 10th Grade Unit 5
MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY
More informationAnonymous Math 361: Homework 5. x i = 1 (1 u i )
Anonymous Mth 36: Homewor 5 Rudin. Let I be the set of ll u (u,..., u ) R with u i for ll i; let Q be the set of ll x (x,..., x ) R with x i, x i. (I is the unit cube; Q is the stndrd simplex in R ). Define
More information