Chapter. Radicals (Surds) Contents: A Radicals on a number line. B Operations with radicals C Expansions with radicals D Division by radicals
|
|
- Liliana Bradley
- 7 years ago
- Views:
Transcription
1 Chter 4 Rdils (Surds) Contents: A Rdils on numer line B Oertions with rdils C Exnsions with rdils D Division y rdils
2 88 RADICALS (SURDS) (Chter 4) INTRODUCTION In revious yers we used the Theorem of Pythgors to find the length of the third side of tringle. Our nswers often involved rdils suh s,,, nd so on. ~` A rdil is numer tht is written using the rdil sign. Rdils suh s 4 nd 9 re rtionl sine 4= nd 9=. Rdils suh s, nd re irrtionl. They hve deiml exnsions whih neither terminte nor reur. Irrtionl rdils re lso known s surds. RESEARCH ² Where did the nmes rdil nd surd ome from? ² Why do we use the word irrtionl to desrie some numers? ² Before we hd lultors nd omuters, finding deiml reresenttions for numers like to four or five deiml les ws uite diffiult nd time onsuming. Imgine hving to find :44 orret to five deiml les using long division! A method ws devised to do this lultion uikly. Wht ws the roess? SQUARE ROOTS The sure root of or is the ositive numer whih oeys the rule =. For to hve mening we reuire to e non-negtive, i.e., > 0. For exmle, = or ( ) =. Note tht A 4=, not, sine the sure root of numer nnot e negtive. RADICALS ON A NUMBER LINE If we onvert rdil suh s to deiml we n find its roximte osition on numer line. ¼ : 0, so is lose to 4. ~` 0
3 RADICALS (SURDS) (Chter 4) 89 We n lso onstrut the osition of on numer line using ruler nd omss. Sine + =( ), we n use right-ngled tringle with sides of length, nd. Ste : Drw numer line nd mrk the numers 0,,, nd on it, m rt. Ste : Ste : Ste 4: EXERCISE 4A With omss oint on, drw n r ove. Do the sme with omss oint on using the sme rdius. Drw the erendiulr t through the intersetion of these rs, nd mrk off m. Cll this oint A. Comlete the right ngled tringle. Its sides re, nd m. With entre O nd rdius OA, drw n r through A to meet the numer line. It meets the numer line t. Notie tht +4 ==( ). Lote on numer line using n urte onstrution. The sum of the sures of whih two ositive integers is? Aurtely onstrut the osition of on numer line. 0 Cn we onstrut the ext osition of on numer line using the method ove? 4 nnot e written s the sum of two sures so the ove method nnot e used for loting 4 ~` on the numer line. However, 4 =, so 4 = +( ). We n thus onstrut right ngled tringle with sides of length 4, nd. Use suh tringle to urtely lote on numer line. ~` DEMO B OPERATIONS WITH RADICALS ADDING AND SUBTRACTING RADICALS We n dd nd sutrt like rdils in the sme wy s we do like terms in lger. For exmle: ² just s + =, + = ² just s 4 =, 4 =. Exmle Simlify: 4 ( ) 4 = = ( ) = + =
4 90 RADICALS (SURDS) (Chter 4) SIMPLIFYING PRODUCTS We hve estlished in revious yers tht: =( ) = = r = Exmle Simlify: ( ) ( ) µ 4 ( ) = = ( ) = = µ 4 = 4 ( ) = =8 Exmle Simlifying: ( ) ( ) ( ) = =9 =8 ( ) = = = 8 Exmle 4 Write in simlest form: 4 With rtie you should not need the middle stes. = = 0 4 = 4 = =
5 RADICALS (SURDS) (Chter 4) 9 Exmle Simlify: = = = = = = 4 = EXERCISE 4B. Simlify: + 8 d + e ( ) f ( + ) g + h + i ( ) j ( + ) + ( ) k ( ) ( ) l ( ) ( ) Simlify: ( ) ( ) ( ) d e ( ) f ( ) g i ( ) j ( ) 4 k Simlify: µ h µ l ( ) (4 ) ( ) d ( ) e ( ) f ( ) g ( ) h ( 0) i ( 0) µ µ µ 0 j 4 k l m ( 4 ) n ( ) o ( ) ( )( ) ( ) r ( ) 4 Simlify: d 9
6 9 RADICALS (SURDS) (Chter 4) Simlify: d e f ( ) g h i j ( ) ( ) k ( ) l ( ) Simlify: 8 8 d 0 e f g 8 h 0 i j 4 k 4 4 l 98 Is 9+ = 9+? Is =? Are + = + nd = ossile lws for rdil numers? 8 Prove tht = for ll ositive numers nd. Hint: Consider ( ) nd ( ). r Prove tht = for > 0 nd >0. SIMPLEST RADICAL FORM A rdil is in simlest form when the numer under the rdil sign is the smllest ossile integer. Exmle Write 8 in simlest form. 8 = 4 = 4 = We look for the lrgest erfet sure tht n e tken out s ftor of this numer. = 4 8= 8 is not in simlest form s 8 n e further simlified into. In simlest form, = 4.
7 RADICALS (SURDS) (Chter 4) 9 Exmle It my e useful Write to do rime 4 in simlest 4 rdil form. = ftoristion of the 4 numer under the = 4 rdil sign. =4 = EXERCISE 4B. Write in the form k where k is n integer: 8 0 d 98 e f 00 g h Write in the form k where k is n integer: 48 d 00 Write in the form k where k Z : d 00 4 Write in simlest rdil form: d e 48 f g 4 h i j 0 k l 000 Write in simlest rdil form + n where, Q, n Z : 4+ 8 e + f g d 8 4 h 00 0 C EXPANSIONS WITH RADICALS The rules for exnding rdil exressions ontining rkets re identil to those for ordinry lger. ( + ) = + ( + )( + d) = + d + + d ( + ) = + + ( + )( ) =
8 94 RADICALS (SURDS) (Chter 4) Exmle 8 Simlify: ( + ) ( ) ( + ) = + ( ) =4+ = + = 4 With rtie you should not need the middle stes. Exmle 9 Exnd nd simlify: ( + ) ( ) ( + ) = + = ( ) = + = + EXERCISE 4C Exnd nd simlify: 4( + ) ( + ) (4 ) d ( 4) e ( + ) f ( ) g ( + ) h ( ) i ( ) j ( ) k ( + ) l ( + + ) Exnd nd simlify: (4 + ) ( ) ( ) d ( + ) e ( ) f ( + ) g ( ) h ( + ) i ( ) j ( +4) k ( ) l ( ) (4 ) Exmle 0 Exnd nd simlify: ( + )( + ) ( + )( ) ( + )( + ) = ( + )+( + ) =+ + + =8+ ( + )( ) =(+ )( + ) = ( + )+( + )( ) =+ =
9 RADICALS (SURDS) (Chter 4) 9 Exnd nd simlify: ( + )( + ) ( + )( + ) ( + )( ) d (4 )( + ) e ( + )( ) f ( )( + ) g ( + )( ) h ( + )( ) i ( + )( +) j ( )( + ) Exmle Exnd nd simlify: ( +) ( ) ( +) =( ) + () + =+ +9 =+ ( ) =( + ) =( ) + ( ) + ( ) = + =8 4 Exnd nd simlify: ( + ) ( +) ( ) d ( + ) e ( ) f (4 ) g ( + ) h ( ) i ( ) j ( +) k ( ) l ( ) Exmle Exnd nd simlify: (4 + )(4 ) ( + )( ) (4 + )(4 ) =4 ( ) = =4 ( + )( ) =( ) =8 9 = Exnd nd simlify: ( + )( ) ( )( +) ( + )( ) d ( 4)( +4) e ( )( +) f ( + )( ) g ( )( + ) h ( + )( )
10 9 RADICALS (SURDS) (Chter 4) i ( + )( ) j ( )( + ) k ( )( + ) l ( + )( ) D DIVISION BY RADICALS In numers like nd 9 we hve divided y rdil. It is ustomry to simlify these numers y rewriting them without the rdil in the denomintor. INVESTIGATION DIVISION BY In this investigtion we onsider frtions of the form where nd re rel numers. To remove the rdil from the denomintor, there re two methods we ould use: ² slitting the numertor ² rtionlising the denomintor Wht to do: Consider the frtion. Sine is ftor of, slit the into. Simlify. Cn the method of slitting the numertor e used to simlify? Consider the frtion. If we multily this frtion y, re we hnging its vlue? Simlify y multilying oth its numertor nd denomintor y. 4 The method in is lled rtionlising the denomintor. Will this method work for ll frtions of the form where nd re rel? From the Investigtion ove, you should hve found tht for ny frtion of the form, we n remove the rdil from the denomintor y multilying y. Sine =, we do not hnge the vlue of the frtion.
11 RADICALS (SURDS) (Chter 4) 9 Exmle Write with n integer denomintor: Multilying the originl numer y or does not hnge its vlue. = = = = = = EXERCISE 4D. Write with integer denomintor: f k g l RADICAL CONJUGATES h 9 d i m n 00 r s e j o t 4 ( ) Rdil exressions suh s + nd whih re identil exet for oosing signs in the middle, re lled rdil onjugtes. The rdil onjugte of + is. INVESTIGATION RADICAL CONJUGATES Frtions of the form + n lso e simlified to remove the rdil from the denomintor. To do this we use rdil onjugtes. Wht to do: Exnd nd simlify: ( + )( ) ( )( +) Wht do you notie out your results in?
12 98 RADICALS (SURDS) (Chter 4) Show tht ( + for ny integers )( nd, the following roduts re integers: ) ( )( + ) 4 Coy nd omlete: To remove the rdils from the denomintor of frtion, we n multily the denomintor y its... Wht must we do to the numertor of the frtion to ensure we do not hnge its vlue? From the Investigtion ove, we should hve found tht: to remove the rdils from the denomintor of frtion, we multily oth the numertor nd the denomintor y the rdil onjugte of the denomintor. Exmle 4 Write 4 with n integer denomintor. µ Ã 4 4 = +! + = 4 9 ( + ) = ( + ) =+ Exmle Write + : with integer denomintor in the form + where, Q. + µ Ã = +! = = = So, = nd =.
13 RADICALS (SURDS) (Chter 4) 99 EXERCISE 4D. Write with integer denomintor: + + d e + f 4 g h + Write in the form + where, Q : 4 + d + Write in the form + where, Q : 4 + d If, nd re integers, show tht ( + )( ) is n integer. Write with n integer denomintor: i + ii. If nd re integers, show tht ( + )( ) is lso n integer. Write with n integer denomintor: i ii + LINKS lik here HOW A CALCULATOR CALCULATES RATIONAL NUMBERS Ares of intertion: Humn ingenuity REVIEW SET 4A Simlify: ( ) µ 4 Coy nd omlete: + =(::::::) Simlify: d 4 Use to urtely onstrut the osition of 0 on numer line using ruler nd omss. d 0
14 00 RADICALS (SURDS) (Chter 4) 4 Write 8 in simlest rdil form. Hene, simlify 8. Write 98 in simlest rdil form. Exnd nd simlify: ( +) ( ) ( + ) d ( ) e ( + )( ) f ( + )( ) Write with n integer denomintor: Write in the form + where, Q : + REVIEW SET 4B Simlify: 8 ( ) d 4 9 Find the ext osition of on numer line using ruler nd omss onstrution. Exlin your method. Hint: Look for two ositive integers nd suh tht =. Simlify: 4 4 Simlify: Write in simlest rdil form: Exnd nd simlify: ( ) ( ) ( ) d ( + ) e ( )( + ) f ( + )( ) Write with integer denomintor: Write in the form + where, Q :
c b 5.00 10 5 N/m 2 (0.120 m 3 0.200 m 3 ), = 4.00 10 4 J. W total = W a b + W b c 2.00
Chter 19, exmle rolems: (19.06) A gs undergoes two roesses. First: onstnt volume @ 0.200 m 3, isohori. Pressure inreses from 2.00 10 5 P to 5.00 10 5 P. Seond: Constnt ressure @ 5.00 10 5 P, isori. olume
More informationIf two triangles are perspective from a point, then they are also perspective from a line.
Mth 487 hter 4 Prtie Prolem Solutions 1. Give the definition of eh of the following terms: () omlete qudrngle omlete qudrngle is set of four oints, no three of whih re olliner, nd the six lines inident
More informationChapter. 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 informationThe 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 informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationChapter. Fractions. Contents: A Representing fractions
Chpter Frtions Contents: A Representing rtions B Frtions o regulr shpes C Equl rtions D Simpliying rtions E Frtions o quntities F Compring rtion sizes G Improper rtions nd mixed numers 08 FRACTIONS (Chpter
More informationAngles 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 informationPolynomial 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 informationModule 5. Three-phase AC Circuits. Version 2 EE IIT, Kharagpur
Module 5 Three-hse A iruits Version EE IIT, Khrgur esson 8 Three-hse Blned Suly Version EE IIT, Khrgur In the module, ontining six lessons (-7), the study of iruits, onsisting of the liner elements resistne,
More informationMATH 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 informationMATH 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 informationWords 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 informationFactoring 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 informationEnd of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.
End of term: TEST A You will need penil nd ruler. Yer Nme Clss Dte Complete the missing numers in the sequenes elow. 8 30 3 28 2 9 25 00 75 25 2 Put irle round ll of the following shpes whih hve 3 shded.
More information5.6 POSITIVE INTEGRAL EXPONENTS
54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section
More informationRatio and Proportion
Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty
More informationUnit 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 informationOperations 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 informationPROF. 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 informationSECTION 7-2 Law of Cosines
516 7 Additionl Topis in Trigonometry h d sin s () tn h h d 50. Surveying. The lyout in the figure t right is used to determine n inessile height h when seline d in plne perpendiulr to h n e estlished
More informationDensity Curve. Continuous Distributions. Continuous Distribution. Density Curve. Meaning of Area Under Curve. Meaning of Area Under Curve
Continuous Distributions Rndom Vribles of the Continuous Tye Density Curve Perent Density funtion f () f() A smooth urve tht fit the distribution 6 7 9 Test sores Density Curve Perent Probbility Density
More informationSPECIAL 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 informationHow To Find The Re Of Tringle
Heron s Formul for Tringulr Are y Christy Willims, Crystl Holom, nd Kyl Gifford Heron of Alexndri Physiist, mthemtiin, nd engineer Tught t the museum in Alexndri Interests were more prtil (mehnis, engineering,
More information1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5.
. Definition, Bsi onepts, Types. Addition nd Sutrtion of Mtries. Slr Multiplition. Assignment nd nswer key. Mtrix Multiplition. Assignment nd nswer key. Determinnt x x (digonl, minors, properties) summry
More informationAppendix 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 informationAlgebra 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 informationRadius of the Earth - Radii Used in Geodesy James R. Clynch Naval Postgraduate School, 2002
dius of the Erth - dii Used in Geodesy Jmes. Clynh vl Postgrdute Shool, 00 I. Three dii of Erth nd Their Use There re three rdii tht ome into use in geodesy. These re funtion of ltitude in the ellipsoidl
More informationWHAT 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 informationAngles and Triangles
nges nd Tringes n nge is formed when two rys hve ommon strting point or vertex. The mesure of n nge is given in degrees, with ompete revoution representing 360 degrees. Some fmiir nges inude nother fmiir
More informationMath 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.
Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose
More informationRIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
More informationMath 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 informationLesson 4.1 Triangle Sum Conjecture
Lesson 4.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q 2., y 3., b 31 82 p 98 q 28 53 y 17 79 23 50 b 4. r, s, 5., y 6. y t t s r 100 85 100 y 30 4 7 y 31 7. s 8.
More informationNQF Level: 2 US No: 7480
NQF Level: 2 US No: 7480 Assessment Guide Primry Agriculture Rtionl nd irrtionl numers nd numer systems Assessor:.......................................... Workplce / Compny:.................................
More information1 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 informationMaximum 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 informationThe Pythagorean Theorem
The Pythgoren Theorem Pythgors ws Greek mthemtiin nd philosopher, orn on the islnd of Smos (. 58 BC). He founded numer of shools, one in prtiulr in town in southern Itly lled Crotone, whose memers eventully
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationP.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 informationHW 9. Problem 14.5. a. To Find:
HW 9 Problem 14.5. To Find: ( The number-verge moleulr weight (b The weight-verge moleulr weight ( The degree of polymeriztion for the given polypropylene mteril Moleulr Weight Rnge (g/mol x i w i 8,000
More informationBinary 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 informationCOMPLEX FRACTIONS. section. Simplifying Complex Fractions
58 (6-6) Chpter 6 Rtionl Epressions undles tht they cn ttch while working together for 0 hours. 00 600 6 FIGURE FOR EXERCISE 9 95. Selling. George sells one gzine suscription every 0 inutes, wheres Theres
More informationPractice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn
Prtie Test 2 1. A highwy urve hs rdius of 0.14 km nd is unnked. A r weighing 12 kn goes round the urve t speed of 24 m/s without slipping. Wht is the mgnitude of the horizontl fore of the rod on the r?
More informationInterior and exterior angles add up to 180. Level 5 exterior angle
22 ngles n proof Ientify interior n exterior ngles in tringles n qurilterls lulte interior n exterior ngles of tringles n qurilterls Unerstn the ie of proof Reognise the ifferene etween onventions, efinitions
More informationSOLVING 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 information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued
More informationOr 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 informationIn order to master the techniques explained here it is vital that you undertake the practice exercises provided.
Tringle formule m-ty-tringleformule-009-1 ommonmthemtilprolemistofindthenglesorlengthsofthesidesoftringlewhen some,utnotllofthesequntitiesreknown.itislsousefultoeletolultethere of tringle from some of
More informationMultiplication 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 informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationVectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m
. Slr prout (ot prout): = osθ Vetors Summry Lws of ot prout: (i) = (ii) ( ) = = (iii) = (ngle etween two ientil vetors is egrees) (iv) = n re perpeniulr Applitions: (i) Projetion vetor: B Length of projetion
More information1. 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 informationQuick 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 informationCypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:
Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationMath Review for Algebra and Precalculus
Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Mt Review for Alger nd Prelulus Stnley Oken Deprtment of Mtemtis
More informationand 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 informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationFUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation
FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does
More informationCS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001
CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic
More informationALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS
CHAPTER ALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS Although people tody re mking greter use of deciml frctions s they work with clcultors, computers, nd the metric system, common
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationSOLVING 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 informationLearning Outcomes. Computer Systems - Architecture Lecture 4 - Boolean Logic. What is Logic? Boolean Logic 10/28/2010
/28/2 Lerning Outcomes At the end of this lecture you should: Computer Systems - Architecture Lecture 4 - Boolen Logic Eddie Edwrds eedwrds@doc.ic.c.uk http://www.doc.ic.c.uk/~eedwrds/compsys (Hevily sed
More informationIndices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková
Indices and Surds The term indices refers to the power to which a number is raised. Thus x is a number with an index of. People prefer the phrase "x to the power of ". Term surds is not often used, instead
More informationPythagorean Triples and Rational Points on the Unit Circle
Pythagorean Triles and Rational Points on the Unit Circle Solutions Below are samle solutions to the roblems osed. You may find that your solutions are different in form and you may have found atterns
More informationGeometry 7-1 Geometric Mean and the Pythagorean Theorem
Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More informationwww.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 informationExample 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 informationCalculating Principal Strains using a Rectangular Strain Gage Rosette
Clulting Prinipl Strins using Retngulr Strin Gge Rosette Strin gge rosettes re used often in engineering prtie to determine strin sttes t speifi points on struture. Figure illustrtes three ommonly used
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x
More informationMA 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 informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More information32. The Tangency Problem of Apollonius.
. The Tngeny olem of Apollonius. Constut ll iles tngent to thee given iles. This eleted polem ws posed y Apollinius of eg (. 60-70 BC), the getest mthemtiin of ntiquity fte Eulid nd Ahimedes. His mjo wok
More information50 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 information1.2 The Integers and Rational Numbers
.2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl
More informationSection 5-4 Trigonometric Functions
5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More information0.1 Basic Set Theory and Interval Notation
0.1 Bsic Set Theory nd Intervl Nottion 3 0.1 Bsic Set Theory nd Intervl Nottion 0.1.1 Some Bsic Set Theory Notions Like ll good Mth ooks, we egin with definition. Definition 0.1. A set is well-defined
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint
More informationVolumes by Cylindrical Shells: the Shell Method
olumes Clinril Shells: the Shell Metho Another metho of fin the volumes of solis of revolution is the shell metho. It n usull fin volumes tht re otherwise iffiult to evlute using the Dis / Wsher metho.
More informationRadius of the Earth - Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
More informationOrthopoles and the Pappus Theorem
Forum Geometriorum Volume 4 (2004) 53 59. FORUM GEOM ISSN 1534-1178 Orthopoles n the Pppus Theorem tul Dixit n Drij Grinerg strt. If the verties of tringle re projete onto given line, the perpeniulrs from
More informationLesson 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 informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationPhysics 2102 Lecture 2. Physics 2102
Physics 10 Jonthn Dowling Physics 10 Lecture Electric Fields Chrles-Augustin de Coulomb (1736-1806) Jnury 17, 07 Version: 1/17/07 Wht re we going to lern? A rod mp Electric chrge Electric force on other
More informationSample Problems. Practice Problems
Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationOUTLINE SYSTEM-ON-CHIP DESIGN. GETTING STARTED WITH VHDL August 31, 2015 GAJSKI S Y-CHART (1983) TOP-DOWN DESIGN (1)
August 31, 2015 GETTING STARTED WITH VHDL 2 Top-down design VHDL history Min elements of VHDL Entities nd rhitetures Signls nd proesses Dt types Configurtions Simultor sis The testenh onept OUTLINE 3 GAJSKI
More informationSolving BAMO Problems
Solving BAMO Problems Tom Dvis tomrdvis@erthlink.net http://www.geometer.org/mthcircles Februry 20, 2000 Abstrct Strtegies for solving problems in the BAMO contest (the By Are Mthemticl Olympid). Only
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationWarm-up for Differential Calculus
Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationMore Properties of Limits: Order of Operations
math 30 day 5: calculating its 6 More Proerties of Limits: Order of Oerations THEOREM 45 (Order of Oerations, Continued) Assume that!a f () L and that m and n are ositive integers Then 5 (Power)!a [ f
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationComplex Conjugation and Polynomial Factorization
Comlex Conjugation and Polynomial Factorization Dave L. Renfro Summer 2004 Central Michigan University I. The Remainder Theorem Let P (x) be a olynomial with comlex coe cients 1 and r be a comlex number.
More information1 GSW IPv4 Addressing
1 For s long s I ve een working with the Internet protools, people hve een sying tht IPv6 will e repling IPv4 in ouple of yers time. While this remins true, it s worth knowing out IPv4 ddresses. Even when
More information