Roots, Radicals, and Complex Numbers

Size: px
Start display at page:

Download "Roots, Radicals, and Complex Numbers"

Transcription

1 Chpter 8 Roots, Rils, Comple Numbers Agel, Itermeite Algebr, 7e Lerig Objetives Workig with squre roots Higher-orer roots; ris tht oti vribles Simplifig ril epressios Agel, Itermeite Algebr, 7e Squre Roots (terms efiitios) Ril epressio: A mthemtil lultio or formul ombiig umbers /or vribles usig sums, ifferees, prouts, quotiets (iluig frtios), epoets, roots, logrithms, trig futios, pretheses, brkets, futios, or other mthemtil opertios. Agel, Itermeite Algebr, 7e

2 Squre Roots (terms efiitios) Ri: The vlue isie the ril sig. The vlue ou wt to tke the root of. Agel, Itermeite Algebr, 7e Squre Roots (terms efiitios) Fiig the squre root of give umber is the reverse proess of squrig umber. Whe fiig the squre root of umber, we re etermiig wht umbers, whe multiplie b themselves, result i the give umber. Emples: If 7, the If 7, the ( 7)( 7) 9 Agel, Itermeite Algebr, 7e 5 Squre Roots (terms efiitios) is re the "squre root of." The is lle the ril sig. The epressio isie the ril sig is lle the ri. The etire epressio, iluig the ril sig ri, is lle the ril epressio. Agel, Itermeite Algebr, 7e 6

3 Squre Roots (terms efiitios) The positive or priipl squre root of positive umber is writte s. The egtive squre root is writte s -. b if b Also, the squre root of 0 is 0, writte 0 0. Note tht the priipl squre root of positive umber,, is the positive umber whose squre equls.. Wheever the term squre root is use i this book, the positive or priipl squre root is met to be use. Agel, Itermeite Algebr, 7e 7 Squre Roots (terms efiitios) The ie tells the root of the epressio. Sie squre roots hve ie of, the ie is geerll ot writte i squre root. mes Emple: 5 5 (sie ) 9 9 (sie ) 6 6 Agel, Itermeite Algebr, 7e 8 Squre Roots (terms efiitios) Squre roots of egtive umbers re ot rel umbers. Squre roots of egtive umbers re lle imgir umbers. 5? There is o umber multiplie b itself tht will give ou 5. Imgir umbers will be isusse lter i this hpter Agel, Itermeite Algebr, 7e 9

4 Squre Roots (terms efiitios) A perfet squre is the squre of turl umber.,, 9, 6, 5, 6 re the first si perfet squres. A rtiol umber is oe tht be writte i the form, where b re itegers, b b 0. Rel umbers tht re ot rtiol umbers re lle irrtiol umbers. As eimls, irrtiol umbers re orepetig, otermitig eimls. Agel, Itermeite Algebr, 7e 0 Perfet Squres Agel, Itermeite Algebr, 7e Approimtig Squre Roots Here re the steps to pproimte squre root:. Pik perfet squre tht is lose to the give umber. Tke its squre root.. Divie the origil umber b this result.. Tke the rithmeti me of the result of I the result of II b ig the two umbers iviig b (this is lso lle ʺtkig vergeʺ).. Divie the origil umber b the result of III. 5. Tke the rithmeti me of the result of III the result of IV. 6. Repet steps IV-VI usig this ew result, util the pproimtio is suffiietl lose. Agel, Itermeite Algebr, 7e

5 Approimtig Squre Roots Emple : Approimte the.. 5 is lose to 5 5. /5.. (5 +.)/.7. / ( )/ / So.69 Agel, Itermeite Algebr, 7e Rtiol, Irrtiol, Imgir Numbers Rtiol umbers: the solutio is iteger. 9 Irrtiol umbers: the solutio is ot iteger. Imgir umbers: 9 Agel, Itermeite Algebr, 7e Agel, Itermeite Algebr, 7e 5

6 Cube Fourth Roots is re the ube root of. is re the fourth root of. b if b b if b 8 sie 8 8 sie ( )( )( ) 8 8 sie 8 Agel, Itermeite Algebr, 7e 6 Eve O Iies Eve Iies The th root of,, where is eve ie is oegtive rel umber, is the oegtive rel umber b suh tht b. 8 sie sie0 00 Agel, Itermeite Algebr, 7e 7 Eve O Iies O Iies The th root of,, where is o ie is rel umber, is the rel umber b suh tht b. 6 sie sie (-) Agel, Itermeite Algebr, 7e 8

7 Cube Fourth Roots Note tht the ube root of positive umber is positive umber the ube root of egtive umber is egtive umber. The ri of fourth root (or eve root) must be oegtive umber for the epressio to be rel umber. Agel, Itermeite Algebr, 7e 9 8. Simplifig Rils Agel, Itermeite Algebr, 7e 0 Defiitios A perfet squre is the squre of turl umber.,, 9, 6, 5, 6 re the first si perfet squres. Vribles with epoets m lso be perfet squres. Emples ilue,( ) ( ). A perfet ube is the ube of turl umber., 8, 7, 6, 5, 6 re the first si perfet ubes. Vribles with epoets m lso be perfet ubes. Emples ilue, ( ) ( ). Agel, Itermeite Algebr, 7e

8 Perfet Powers This ie be epe to perfet powers of vrible for ri. The ri is perfet power whe is multiple of the ie of the ri. A quik w to etermie if ri is perfet power for ie is to etermie if the epoet is ivisible b the ie of the ril. Emple: 5 0 Sie the epoet, 0, is ivisible b the ie, 5, 0 is perfet fifth power. Agel, Itermeite Algebr, 7e Prout Rule for Rils For oegtive rel umbers b b b, Emples: Agel, Itermeite Algebr, 7e Prout Rule for Rils To Simplif Rils Usig the Prout Rule. If the ri otis oeffiiet other th, write it s prout of the two umbers, oe of whih is the lrgest perfet power for the ie.. Write eh vrible ftor s prout of two ftors, oe of whih hihis the lrgest perfet power of fthe vrible ibl for the ie.. Use the prout rule to write the ril epressio s prout of rils. Ple ll the perfet powers uer the sme ril.. Simplif the ril otiig the perfet powers. Agel, Itermeite Algebr, 7e

9 Prout Rule for Rils Emples: b b b b b b b *Whe the ril is simplifie, the ri oes ot hve vrible with epoet greter th or equl to the ie. Agel, Itermeite Algebr, 7e 5 Quotiet Rule for Rils For oegtive rel umbers, b 0 b b b, Emples: Simplif ri, if possible Agel, Itermeite Algebr, 7e 6 Quotiet Rule for Rils More Emples: b 6 b 8 8 6b 8 6b 8 6b b 8 8 Agel, Itermeite Algebr, 7e 7

10 8. 5 Aig, Subtrtig, Multiplig Rils Agel, Itermeite Algebr, 7e 8 Like Rils Like rils re rils hvig the sme ris. The re e the sme w like terms re e. Emple: z + 0 z 5 z 8 z Cot be simplifie further. Agel, Itermeite Algebr, 7e 9 Aig & Subtrtig To A or Subtrt Rils. Simplif eh ril epressio.. Combie like rils (if there re ). Emples: (6 ) Agel, Itermeite Algebr, 7e 0

11 CAUTION! The prout rule oes ot ppl to itio or subtrtio! b b + b + b Agel, Itermeite Algebr, 7e Multiplig Rils Multipl: ( 5 ) 5 (8 + 5)(6 ) Use the FOIL metho. ( + 6)( 6) ( ) Notie tht the ier outer terms el. Agel, Itermeite Algebr, 7e Multiplig Rils More Emples: 8 ( 7 ) z z z z z z z 7 5 Agel, Itermeite Algebr, 7e

12 Diviig Rils Agel, Itermeite Algebr, 7e g Rtiolizig Deomitors Emples: To Rtiolize Deomitor 6 Multipl both the umertor the eomitor of the frtio b ril tht will result i the ri i the eomitor beomig perfet power. C t b i lifi f th Agel, Itermeite Algebr, 7e 5 Emples: r pr q r r pq r r pq r r r pq r pq Cot be simplifie further ) ( Simplifig Rils Simplif b rtiolizig the eomitor: 5 + Agel, Itermeite Algebr, 7e ) )( ( ) )( ( + +

13 Simplifig Rils A Ril Epressio is Simplifie Whe the Followig Are All True. No perfet powers re ftors of the ri ll epoets i the ri re less th the ie.. No ri otis frtio.. No eomitor otis ril. Agel, Itermeite Algebr, 7e 7 Simplifig Rils Simplif: ( r + ) 6 5 ( r + ) 5 5 ( r + ) ( r + ) 6 5 ( 5 6) ( 5 ) ( 5 6) ( 0 6) 5 6 ( r + ) ( r + ) ( r + ) 5 6 ( r + ) Agel, Itermeite Algebr, 7e Solvig Ril Equtios Agel, Itermeite Algebr, 7e 9

14 Ril Equtios A ril equtio is equtio tht otis vrible i ri To solve ril equtios suh s these, both sies of the equtio re squre ( ) ( + ) ( ) ( 7) 0 ; 7 ( ) Agel, Itermeite Algebr, 7e 0 Etreous Roots I the previous emple, etreous root ws obtie whe both sies were squre. A etreous root is ot solutio to the origil equtio. Alws hek ll of our solutios ito the origil equtio. 0 ; 7 Chek: Chek: FALSE! Agel, Itermeite Algebr, 7e Two Squre Root Terms To solve equtios with two squre root terms, rewrite the equtio, if eessr so tht there is ol oe term otiig squre root o eh sie of the equtio. Solve the equtio: 5 + ( 5) ( + ) Chek: 7 (7) Agel, Itermeite Algebr, 7e

15 Noril Terms Solve the equtio: b b + ( b ) ( b + ) b 6 8 b + + b + b 0 8 b + ( b 0 ) ( 8 b + ) b + 0b (b + ) b + 0b b + 6 b 88b ( b 8 )( b ) 0 b 8, b Chek: b 8 b b Not solutio. b + 9 Agel, Itermeite Algebr, 7e Summr To Solve Ril Equtios. Rewrite the equtio so tht oe ril otiig vrible is isolte o oe sie of the equtio.. Rise eh sie of the equtio to power equl to the ie of the ril.. Combie like terms.. If the equtio still otis term with vrible i ri, repet steps through. 5. Solve the resultig equtio for the vrible. 6. Chek ll solutios i the origil equtio for etreous solutios. Agel, Itermeite Algebr, 7e 8.7 Rtiol Epoets Agel, Itermeite Algebr, 7e 5

16 Chgig Ril Epressio A ril epressio be writte usig epoets b usig the followig proeure: Whe is oegtive, be ie. Whe is egtive, must be o ( ) ( 7 ) 9 + 7z + z Agel, Itermeite Algebr, 7e 6 Chgig Ril Epressio Whe is oegtive, be ie. Whe is egtive, must be o. Epoetil epressios be overte to ril epressios b reversig the proeure. 5 b b 5 Agel, Itermeite Algebr, 7e 7 Simplifig Ril Epressios This rule be epe so tht rils of the m form be writte s epoetil epressios. For oegtive umber, itegers m, Power m m ( ) m Ie b b ( 8 9) ( ) Agel, Itermeite Algebr, 7e 8

17 Rules of Epoets The rules of epoets from Setio 5. lso ppl whe the epoets re rtiol umbers. For ll rel umbers b ll rtiol umbers m, Prout rule: Quotiet rule: Negtive epoet rule: m m + m m m, 0, 0 m Agel, Itermeite Algebr, 7e 9 Rules of Epoets For ll rel umbers b ll rtiol umbers m, Zero epoet rule: Risig power to power: Risig prout to power : Risig quotiet to power : 0, 0 m m ( ) m m m ( b ) b b m b m m, b 0 Agel, Itermeite Algebr, 7e 50 Rules of Epoets Emples:.) Simplif -/ -/5. 5 (- ) ( 5) ( 5 0) ( 0) ) Simplif ( -/5 ) /. ( 5)( ) ( ) ) Multipl -/9 ( /9 ). ( 9) + ( 9) ( 9) Agel, Itermeite Algebr, 7e 5

18 Ftorig Epressios Emples:.) Ftor / 5/. The smllest of the two epoets is /. / 5/ / ( 5/-(/) ) / ( / ) / ( ) Origil epoet Epoet ftore out.) Ftor -/ + /. The smllest of the two epoets is -/. + -/ + / -/ ( /-(-/) ) -/ ( ) Origil Epoet epoet ftore out Agel, Itermeite Algebr, 7e 5

Chapter 3 Section 3 Lesson Additional Rules for Exponents

Chapter 3 Section 3 Lesson Additional Rules for Exponents Chpter Sectio Lesso Additiol Rules for Epoets Itroductio I this lesso we ll eie soe dditiol rules tht gover the behvior of epoets The rules should be eorized; they will be used ofte i the reiig chpters

More information

Repeated multiplication is represented using exponential notation, for example:

Repeated multiplication is represented using exponential notation, for example: Appedix A: The Lws of Expoets Expoets re short-hd ottio used to represet my fctors multiplied together All of the rules for mipultig expoets my be deduced from the lws of multiplictio d divisio tht you

More information

MATH 90 CHAPTER 5 Name:.

MATH 90 CHAPTER 5 Name:. MATH 90 CHAPTER 5 Nme:. 5.1 Multiplictio of Expoets Need To Kow Recll expoets The ide of expoet properties Apply expoet properties Expoets Expoets me repeted multiplictio. 3 4 3 4 4 ( ) Expoet Properties

More information

Arithmetic Sequences

Arithmetic Sequences Arithmetic equeces A simple wy to geerte sequece is to strt with umber, d dd to it fixed costt d, over d over gi. This type of sequece is clled rithmetic sequece. Defiitio: A rithmetic sequece is sequece

More information

Tallahassee Community College. Simplifying Radicals

Tallahassee Community College. Simplifying Radicals Tllhssee Communit College Simplifing Rdils The squre root of n positive numer is the numer tht n e squred to get the numer whose squre root we re seeking. For emple, 1 euse if we squre we get 1, whih is

More information

Chapter 04.05 System of Equations

Chapter 04.05 System of Equations hpter 04.05 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee

More information

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL - INDICES, LOGARITHMS AND FUNCTION This is the oe of series of bsic tutorils i mthemtics imed t begiers or yoe wtig to refresh themselves o fudmetls.

More information

A function f whose domain is the set of positive integers is called a sequence. The values

A function f whose domain is the set of positive integers is called a sequence. The values EQUENCE: A fuctio f whose domi is the set of positive itegers is clled sequece The vlues f ( ), f (), f (),, f (), re clled the terms of the sequece; f() is the first term, f() is the secod term, f() is

More information

MATH PLACEMENT REVIEW GUIDE

MATH PLACEMENT REVIEW GUIDE MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your

More information

STUDY COURSE BACHELOR OF BUSINESS ADMINISTRATION (B.A.)

STUDY COURSE BACHELOR OF BUSINESS ADMINISTRATION (B.A.) STUDY COURSE BACHELOR OF BUSINESS ADMINISTRATION (B.A. MATHEMATICS (ENGLISH & GERMAN REPETITORIUM 0/06 Prof. Dr. Philipp E. Zeh Mthemtis Prof. Dr. Philipp E. Zeh LITERATURE (GERMAN Böker, F., Formelsmmlug

More information

UNIT FIVE DETERMINANTS

UNIT FIVE DETERMINANTS UNIT FIVE DETERMINANTS. INTRODUTION I uit oe the determit of mtrix ws itroduced d used i the evlutio of cross product. I this chpter we exted the defiitio of determit to y size squre mtrix. The determit

More information

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

Math 114- Intermediate Algebra Integral Exponents & Fractional Exponents (10 )

Math 114- Intermediate Algebra Integral Exponents & Fractional Exponents (10 ) Math 4 Math 4- Itermediate Algebra Itegral Epoets & Fractioal Epoets (0 ) Epoetial Fuctios Epoetial Fuctios ad Graphs I. Epoetial Fuctios The fuctio f ( ) a, where is a real umber, a 0, ad a, is called

More information

Released Assessment Questions, 2015 QUESTIONS

Released Assessment Questions, 2015 QUESTIONS Relesed Assessmet Questios, 15 QUESTIONS Grde 9 Assessmet of Mthemtis Ademi Red the istrutios elow. Alog with this ooklet, mke sure you hve the Aswer Booklet d the Formul Sheet. You my use y spe i this

More information

Calculus Cheat Sheet. except we make f ( x ) arbitrarily large and. Relationship between the limit and one-sided limits

Calculus Cheat Sheet. except we make f ( x ) arbitrarily large and. Relationship between the limit and one-sided limits Clulus Chet Sheet Limits Deiitios Preise Deiitio : We sy lim L i or every ε > 0 there is δ > 0 suh tht wheever 0 δ L < ε. < < the Workig Deiitio : We sy lim L i we mke ( ) s lose to L s we wt y tkig suiietly

More information

Calculus Cheat Sheet. except we make f ( x ) arbitrarily large and. Relationship between the limit and one-sided limits

Calculus Cheat Sheet. except we make f ( x ) arbitrarily large and. Relationship between the limit and one-sided limits Clulus Chet Sheet Limits Deiitios Preise Deiitio : We sy lim ( ) L i or every e > 0 there is > 0 suh tht wheever 0 L < e. < < the Workig Deiitio : We sy lim L i we mke ( ) s lose to L s we wt y tkig suiietly

More information

Section 6.1 Radicals and Rational Exponents

Section 6.1 Radicals and Rational Exponents Sectio 6.1 Radicals ad Ratioal Expoets Defiitio of Square Root The umber b is a square root of a if b The priciple square root of a positive umber is its positive square root ad we deote this root by usig

More information

SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 1355 - INTERMEDIATE ALGEBRA I (3 CREDIT HOURS)

SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 1355 - INTERMEDIATE ALGEBRA I (3 CREDIT HOURS) SINCLAIR COMMUNITY COLLEGE DAYTON OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 1355 - INTERMEDIATE ALGEBRA I (3 CREDIT HOURS) 1. COURSE DESCRIPTION: Ftorig; opertios with polyoils d rtiol expressios; solvig

More information

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is 0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Solving Logarithms and Exponential Equations

Solving Logarithms and Exponential Equations Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:

More information

MATH 181-Exponents and Radicals ( 8 )

MATH 181-Exponents and Radicals ( 8 ) Mth 8 S. Numkr MATH 8-Epots d Rdicls ( 8 ) Itgrl Epots & Frctiol Epots Epotil Fuctios Epotil Fuctios d Grphs I. Epotil Fuctios Th fuctio f ( ), whr is rl umr, 0, d, is clld th potil fuctio, s. Rquirig

More information

Laws of Exponents Learning Strategies

Laws of Exponents Learning Strategies Laws of Epoets Learig Strategies What should studets be able to do withi this iteractive? Studets should be able to uderstad ad use of the laws of epoets. Studets should be able to simplify epressios that

More information

GRADE 4. Fractions WORKSHEETS

GRADE 4. Fractions WORKSHEETS GRADE Frtions WORKSHEETS Types of frtions equivlent frtions This frtion wll shows frtions tht re equivlent. Equivlent frtions re frtions tht re the sme mount. How mny equivlent frtions n you fin? Lel eh

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

Swelling and Mechanical Properties of Hydrogels Composed of. Binary Blends of Inter-linked ph-responsive Microgel Particles

Swelling and Mechanical Properties of Hydrogels Composed of. Binary Blends of Inter-linked ph-responsive Microgel Particles Eletroi Supplemetry Mteril (ESI) for Soft Mtter. This jourl is The Royl Soiety of Chemistry 205 SUPPLEMENTARY INFRMATIN Swellig Mehil Properties of Hyrogels Compose of Biry Bles of Iter-like ph-resposive

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

We will begin this chapter with a quick refresher of what an exponent is.

We will begin this chapter with a quick refresher of what an exponent is. .1 Exoets We will egi this chter with quick refresher of wht exoet is. Recll: So, exoet is how we rereset reeted ultilictio. We wt to tke closer look t the exoet. We will egi with wht the roerties re for

More information

α. Figure 1(iii) shows the inertia force and

α. Figure 1(iii) shows the inertia force and CHPTER DYNMIC ORCE NLYSIS Whe the ierti fores re osiere i the lysis of the mehism, the lysis is kow s ymi fore lysis. Now pplyig D lemert priiple oe my reue ymi system ito equivlet stti system use the

More information

Laws of Exponents. net effect is to multiply with 2 a total of 3 + 5 = 8 times

Laws of Exponents. net effect is to multiply with 2 a total of 3 + 5 = 8 times The Mathematis 11 Competey Test Laws of Expoets (i) multipliatio of two powers: multiply by five times 3 x = ( x x ) x ( x x x x ) = 8 multiply by three times et effet is to multiply with a total of 3

More information

WHAT HAPPENS WHEN YOU MIX COMPLEX NUMBERS WITH PRIME NUMBERS?

WHAT HAPPENS WHEN YOU MIX COMPLEX NUMBERS WITH PRIME NUMBERS? WHAT HAPPES WHE YOU MIX COMPLEX UMBERS WITH PRIME UMBERS? There s n ol syng, you n t pples n ornges. Mthemtns hte n t; they love to throw pples n ornges nto foo proessor n see wht hppens. Sometmes they

More information

Unit 6: Exponents and Radicals

Unit 6: Exponents and Radicals Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

More information

m n Use technology to discover the rules for forms such as a a, various integer values of m and n and a fixed integer value a.

m n Use technology to discover the rules for forms such as a a, various integer values of m and n and a fixed integer value a. TIth.co Alger Expoet Rules ID: 988 Tie required 25 iutes Activity Overview This ctivity llows studets to work idepedetly to discover rules for workig with expoets, such s Multiplictio d Divisio of Like

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

1 Fractions from an advanced point of view

1 Fractions from an advanced point of view 1 Frtions from n vne point of view We re going to stuy frtions from the viewpoint of moern lger, or strt lger. Our gol is to evelop eeper unerstning of wht n men. One onsequene of our eeper unerstning

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example

Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example 2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel

More information

Section 8.3 : De Moivre s Theorem and Applications

Section 8.3 : De Moivre s Theorem and Applications The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =

More information

Complex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have

Complex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have Comple Numbers I spite of Calvi s discomfiture, imagiar umbers (a subset of the set of comple umbers) eist ad are ivaluable i mathematics, egieerig, ad sciece. I fact, i certai fields, such as electrical

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent! MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more

More information

How to Graphically Interpret the Complex Roots of a Quadratic Equation

How to Graphically Interpret the Complex Roots of a Quadratic Equation Universit of Nersk - Linoln DigitlCommons@Universit of Nersk - Linoln MAT Em Epositor Ppers Mth in the Middle Institute Prtnership 7-007 How to Grphill Interpret the Comple Roots of Qudrti Eqution Crmen

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

Polynomials. Common Mistakes

Polynomials. Common Mistakes Polnomils Polnomils Definition A polnomil is single term or sum or difference of terms in which ll vribles hve whole-number eponents nd no vrible ppers in the denomintor. Ech term cn be either constnt,

More information

REVISION SHEET FP2 (AQA) CALCULUS. x x π π. Standard Calculus of Inverse Trig and Hyperbolic Trig Functions = + arcsin x = + ar sinh x

REVISION SHEET FP2 (AQA) CALCULUS. x x π π. Standard Calculus of Inverse Trig and Hyperbolic Trig Functions = + arcsin x = + ar sinh x the Further Mathematics etwork www.fmetwork.org.uk V 07 REVISION SHEET FP (AQA) CALCULUS The mai ideas are: Calculus usig iverse trig fuctios & hperbolic trig fuctios ad their iverses. Calculatig arc legths.

More information

Square & Square Roots

Square & Square Roots Squre & Squre Roots Squre : If nuber is ultiplied by itself then the product is the squre of the nuber. Thus the squre of is x = eg. x x Squre root: The squre root of nuber is one of two equl fctors which

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

50 MATHCOUNTS LECTURES (10) RATIOS, RATES, AND PROPORTIONS

50 MATHCOUNTS LECTURES (10) RATIOS, RATES, AND PROPORTIONS 0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS BASIC KNOWLEDGE () RATIOS: Rtios re use to ompre two or more numers For n two numers n ( 0), the rtio is written s : = / Emple : If 4 stuents in

More information

Chapter. Contents: A Constructing decimal numbers

Chapter. Contents: A Constructing decimal numbers Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting

More information

State the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127

State the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127 ngles 2 CHTER 2.1 Tringles Drw tringle on pper nd lel its ngles, nd. Ter off its orners. Fit ngles, nd together. They mke stright line. This shows tht the ngles in this tringle dd up to 180 ut it is not

More information

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

More information

Words Symbols Diagram. abcde. a + b + c + d + e

Words Symbols Diagram. abcde. a + b + c + d + e Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To

More information

The second difference is the sequence of differences of the first difference sequence, 2

The second difference is the sequence of differences of the first difference sequence, 2 Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

Equation of a line. Line in coordinate geometry. Slope-intercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Point-slope form ( 點 斜 式 )

Equation of a line. Line in coordinate geometry. Slope-intercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Point-slope form ( 點 斜 式 ) Chapter : Liear Equatios Chapter Liear Equatios Lie i coordiate geometr I Cartesia coordiate sstems ( 卡 笛 兒 坐 標 系 統 ), a lie ca be represeted b a liear equatio, i.e., a polomial with degree. But before

More information

Application: Volume. 6.1 Overture. Cylinders

Application: Volume. 6.1 Overture. Cylinders Applictio: Volume 61 Overture I this chpter we preset other pplictio of the defiite itegrl, this time to fid volumes of certi solids As importt s this prticulr pplictio is, more importt is to recogize

More information

Quick Guide to Lisp Implementation

Quick Guide to Lisp Implementation isp Implementtion Hndout Pge 1 o 10 Quik Guide to isp Implementtion Representtion o si dt strutures isp dt strutures re lled S-epressions. The representtion o n S-epression n e roken into two piees, the

More information

Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.

Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function. Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while

More information

The remaining two sides of the right triangle are called the legs of the right triangle.

The remaining two sides of the right triangle are called the legs of the right triangle. 10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right

More information

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

More information

Gray level image enhancement using the Bernstein polynomials

Gray level image enhancement using the Bernstein polynomials Buletiul Ştiiţiic l Uiersităţii "Politehic" di Timişor Seri ELECTRONICĂ şi TELECOMUNICAŢII TRANSACTIONS o ELECTRONICS d COMMUNICATIONS Tom 47(6), Fscicol -, 00 Gry leel imge ehcemet usig the Berstei polyomils

More information

MATHEMATICS SYLLABUS SECONDARY 7th YEAR

MATHEMATICS SYLLABUS SECONDARY 7th YEAR Europe Schools Office of the Secretry-Geerl Pedgogicl developmet Uit Ref.: 2011-01-D-41-e-2 Orig.: DE MATHEMATICS SYLLABUS SECONDARY 7th YEAR Stdrd level 5 period/week course Approved y the Joit Techig

More information

The Euler Totient, the Möbius and the Divisor Functions

The Euler Totient, the Möbius and the Divisor Functions The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship

More information

8.1 Arithmetic Sequences

8.1 Arithmetic Sequences MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first

More information

SOLVING QUADRATIC EQUATIONS BY FACTORING

SOLVING QUADRATIC EQUATIONS BY FACTORING 6.6 Solving Qudrti Equtions y Ftoring (6 31) 307 In this setion The Zero Ftor Property Applitions 6.6 SOLVING QUADRATIC EQUATIONS BY FACTORING The tehniques of ftoring n e used to solve equtions involving

More information

n Using the formula we get a confidence interval of 80±1.64

n Using the formula we get a confidence interval of 80±1.64 9.52 The professor of sttistics oticed tht the rks i his course re orlly distributed. He hs lso oticed tht his orig clss verge is 73% with stdrd devitio of 12% o their fil exs. His fteroo clsses verge

More information

Maximum area of polygon

Maximum area of polygon Mimum re of polygon Suppose I give you n stiks. They might e of ifferent lengths, or the sme length, or some the sme s others, et. Now there re lots of polygons you n form with those stiks. Your jo is

More information

ASSOCIATION AND EFFECT. Nils Toft, Jens Frederik Agger and Jeanett Bruun

ASSOCIATION AND EFFECT. Nils Toft, Jens Frederik Agger and Jeanett Bruun Smple hpter from Itrodutio to Veteriry Epidemiology Biofoli, 2004 7 MEASURES OF ASSOCIATION AND EFFECT 7.1 Itrodutio Nils Toft, Jes Frederik Agger d Jeett Bruu Ofte our oer i epidemiologil study is to

More information

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values) www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input

More information

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes. LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.

More information

1. Area under a curve region bounded by the given function, vertical lines and the x axis.

1. Area under a curve region bounded by the given function, vertical lines and the x axis. Ares y Integrtion. Are uner urve region oune y the given funtion, vertil lines n the is.. Are uner urve region oune y the given funtion, horizontl lines n the y is.. Are etween urves efine y two given

More information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define

More information

SOLVING EQUATIONS BY FACTORING

SOLVING EQUATIONS BY FACTORING 316 (5-60) Chpter 5 Exponents nd Polynomils 5.9 SOLVING EQUATIONS BY FACTORING In this setion The Zero Ftor Property Applitions helpful hint Note tht the zero ftor property is our seond exmple of getting

More information

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

2-3 The Remainder and Factor Theorems

2-3 The Remainder and Factor Theorems - The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

MATHEMATICAL INDUCTION

MATHEMATICAL INDUCTION MATHEMATICAL INDUCTION. Itroductio Mthemtics distiguishes itself from the other scieces i tht it is built upo set of xioms d defiitios, o which ll subsequet theorems rely. All theorems c be derived, or

More information

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right. Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

3. Greatest Common Divisor - Least Common Multiple

3. Greatest Common Divisor - Least Common Multiple 3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

More information

Infinite Sequences and Series

Infinite Sequences and Series CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

More information

Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

More information

x(x 1)(x 2)... (x k + 1) = [x] k n+m 1

x(x 1)(x 2)... (x k + 1) = [x] k n+m 1 1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,

More information

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

More information

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

More information

Authorized licensed use limited to: University of Illinois. Downloaded on July 27,2010 at 06:52:39 UTC from IEEE Xplore. Restrictions apply.

Authorized licensed use limited to: University of Illinois. Downloaded on July 27,2010 at 06:52:39 UTC from IEEE Xplore. Restrictions apply. Uiversl Dt Compressio d Lier Predictio Meir Feder d Adrew C. Siger y Jury, 998 The reltioship betwee predictio d dt compressio c be exteded to uiversl predictio schemes d uiversl dt compressio. Recet work

More information

OxCORT v4 Quick Guide Revision Class Reports

OxCORT v4 Quick Guide Revision Class Reports OxCORT v4 Quik Guie Revision Clss Reports This quik guie is suitble for the following roles: Tutor This quik guie reltes to the following menu options: Crete Revision Clss Reports pg 1 Crete Revision Clss

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern. 5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

Lesson 2.1 Inductive Reasoning

Lesson 2.1 Inductive Reasoning Lesson.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 1, 16,,. 400, 00, 100, 0,,,. 1 8, 7, 1, 4,, 4.,,, 1, 1, 0,,. 60, 180, 10,

More information

A. Description: A simple queueing system is shown in Fig. 16-1. Customers arrive randomly at an average rate of

A. Description: A simple queueing system is shown in Fig. 16-1. Customers arrive randomly at an average rate of Queueig Theory INTRODUCTION Queueig theory dels with the study of queues (witig lies). Queues boud i rcticl situtios. The erliest use of queueig theory ws i the desig of telehoe system. Alictios of queueig

More information

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean 1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

More information

Introduction to Hypothesis Testing

Introduction to Hypothesis Testing Itroductio to Hypothesis Testig I Cosumer Reports, April, 978, the results of tste test were reported. Cosumer Reports commeted, "we do't cosider this result to be sttisticlly sigifict." At the time, Miller

More information