Tallahassee Community College. Simplifying Radicals

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Tallahassee Community College. Simplifying Radicals"

Transcription

1 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 the numer whose squre root is eing found. 8 euse euse The smol is lled rdil nd it is red s the squre root of. The numer underneth the rdil is lled the rdind. In the epression., the rdind is It should e noted tht eh positive numer hs squre roots. One is the positive or prinipl squre root, nd the other is the negtive squre root. euse euse ( ) We re usull interested in the positive squre root. If we wnt the negtive root, we put negtive sign in front of the rdil. 7 Note tht we nnot hve negtive sign under the rdil. is not rel numer, euse there is no numer tht we n multipl itself nd get. (In MAT 10 nd MAC 110 ou will lern how to del with this sitution.)

2 To simplif rdil we must look for nd remove n perfet squre ftors tht m e in the rdind. REMEMBER tht perfet squre is the squre of n integer. 1 Perfet squres 8 A rdil epression is in simplest form if the rdind ontins no perfet squre ftors. To simplif rdil we will first find the prime ftoriztion of the rdind nd rewrite the rdind in eponentil form. prime ftoriztion of in eponentil form. If the eponent is n even numer, then the numer itself is perfet squre. To tke the squre root of EXAMPLE: we remove the rdil nd divide the eponent. / 8 One we hve divided the eponent, we n multipl out the remining ftors. EXAMPLES: 81 / 1 / / / 7 / 7 1 Often the numer we wish to simplif is not perfet squre. We then hve to find n perfet squre ftors ontined in the numer nd remove them from under the rdil tking their squre roots. To simplif 0, first find the prime ftoriztion of 0. 0 NOTICE tht the eponents re odd numers. This mens tht nd 1 re not perfet squres.

3 An prime ftor with n eponent of 1 will not e perfet squre nor will it ontin perfet squre. An prime ftor with n even eponent will e perfet squre. An prime ftor with n odd eponent of or higher will ontin perfet squre ftor ontins perfet squre is written s 1 The Produt Propert of Squre Roots llows us to rewrite produt under rdil s produt of seprte rdils We now hve on rdil whih is perfet squre nd one whih is not. We n tke the squre root of the perfet squre nd multipl the ftors remining under the rdil. 10 The omplete proess is s follows: Find the prime ftoriztion of 0 s Seprte the perfet squres Tke squre roots. The solution is red s times the squre root of 10 EXAMPLE: Simplif / 1 1 Find the prime ftoriztion of s Seprte the perfet squres Tke squre roots. Simplif to get The solution is red s times the squre root of

4 EXAMPLE: Simplif 180. Notie tht this is times the squre root of 180. We must simplif 180 first nd then multipl. 180 Find the prime ftoriztion of 180 Seprte the perfet squres / / Tke squre roots. Multipl 0 The solution is red s 0 times the squre root of EXAMPLE: Simplif / Find the prime ftoriztion of s Seprte the perfet squres Tke squre roots. Simplif to get nd multipl The solution is red s times the squre root of Mn of the epressions we will need to simplif will ontin vriles. euse ( ) euse ( ) An vrile rdil epression whih hs n even eponent will e perfet squre. An vrile rdil epression whih hs n eponent of 1 will not e perfet squre nor will it ontin perfet squre ftor. An vrile epression whih hs n odd eponent of or higher will ontin perfet squre ftor. /

5 EXAMPLES: / 18 / 18 8 / 8 We often hve rdils whih hve oth numers nd vriles. EXAMPLE: Simplif EXAMPLE: Simplif EXAMPLE: Simplif 7 7 Find the prime ftoriztion of s Seprte the perfet squres nd tke squre roots Find the prime ftoriztion of 7,, nd Seprte the perfet squres nd tke squre roots Multipl the numers Find the prime ftoriztion of 7 nd Seprte the perfet squres nd tke squre roots Multipl the numers nd multipl the vriles

6 EXAMPLE: Simplif Find the prime ftoriztion of 18 s Seprte the perfet squres nd tke squre roots Multipl the numers nd multipl the vriles EXERCISES. Simplif eh of the following. ) ) 1 ) 0 d) 80 e) 1 f) 1 g) h) 0 i) 8 10 j) 10 KEY ) 7 ) ) 10 d) 1 e) g) j) 0 7 f) 7 h) i) 1

11.1 Conic sections (conics)

11.1 Conic sections (conics) . Coni setions onis Coni setions re formed the intersetion of plne with right irulr one. The tpe of the urve depends on the ngle t whih the plne intersets the surfe A irle ws studied in lger in se.. We

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

Simple Nonlinear Graphs

Simple Nonlinear Graphs Simple Nonliner Grphs Curriulum Re www.mthletis.om Simple SIMPLE Nonliner NONLINEAR Grphs GRAPHS Liner equtions hve the form = m+ where the power of (n ) is lws. The re lle Liner euse their grphs re stright

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

An Insight into Quadratic Equations and Cubic Equations with Real Coefficients

An Insight into Quadratic Equations and Cubic Equations with Real Coefficients An Insight into Qurti Equtions n Cubi Equtions with Rel Coeffiients Qurti Equtions A qurti eqution is n eqution of the form x + bx + =, where o It n be solve quikly if we n ftorize the expression x + bx

More information

5.6 POSITIVE INTEGRAL EXPONENTS

5.6 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 information

Pre-algebra 7* In your group consider the following problems:

Pre-algebra 7* In your group consider the following problems: Pre-lger * Group Activit # Group Memers: In our group consider the following prolems: 1) If ever person in the room, including the techer, were to shke hnds with ever other person ectl one time, how mn

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

The AVL Tree Rotations Tutorial

The AVL Tree Rotations Tutorial The AVL Tree Rottions Tutoril By John Hrgrove Version 1.0.1, Updted Mr-22-2007 Astrt I wrote this doument in n effort to over wht I onsider to e drk re of the AVL Tree onept. When presented with the tsk

More information

Chapter. Radicals (Surds) Contents: A Radicals on a number line. B Operations with radicals C Expansions with radicals D Division by radicals

Chapter. Radicals (Surds) Contents: A Radicals on a number line. B Operations with radicals C Expansions with radicals D Division by radicals Chter 4 Rdils (Surds) Contents: A Rdils on numer line B Oertions with rdils C Exnsions with rdils D Division y rdils 88 RADICALS (SURDS) (Chter 4) INTRODUCTION In revious yers we used the Theorem of Pythgors

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

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

10.3 Systems of Linear Equations: Determinants

10.3 Systems of Linear Equations: Determinants 758 CHAPTER 10 Systems of Equtions nd Inequlities 10.3 Systems of Liner Equtions: Determinnts OBJECTIVES 1 Evlute 2 y 2 Determinnts 2 Use Crmer s Rule to Solve System of Two Equtions Contining Two Vriles

More information

Chapter 9: Quadratic Equations

Chapter 9: Quadratic Equations Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.

More information

Quadratic Equations - 1

Quadratic Equations - 1 Alger Module A60 Qudrtic Equtions - 1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions - 1 Sttement of Prerequisite

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

The area of the larger square is: IF it s a right triangle, THEN + =

The area of the larger square is: IF it s a right triangle, THEN + = 8.1 Pythgoren Theorem nd 2D Applitions The Pythgoren Theorem sttes tht IF tringle is right tringle, THEN the sum of the squres of the lengths of the legs equls the squre of the hypotenuse lengths. Tht

More information

D e c i m a l s DECIMALS.

D e c i m a l s DECIMALS. D e i m l s DECIMALS www.mthletis.om.u Deimls DECIMALS A deiml numer is sed on ple vlue. 214.84 hs 2 hundreds, 1 ten, 4 units, 8 tenths nd 4 hundredths. Sometimes different 'levels' of ple vlue re needed

More information

SIMPLIFYING SQUARE ROOTS EXAMPLES

SIMPLIFYING SQUARE ROOTS EXAMPLES SIMPLIFYING SQUARE ROOTS EXAMPLES 1. Definition of a simplified form for a square root The square root of a positive integer is in simplest form if the radicand has no perfect square factor other than

More information

THE PYTHAGOREAN THEOREM

THE PYTHAGOREAN THEOREM THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this

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

Chapter 6: Radical Functions and Rational Exponents

Chapter 6: Radical Functions and Rational Exponents Algebra B: Chapter 6 Notes 1 Chapter 6: Radical Functions and Rational Eponents Concept Bte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Propert of Eponents:

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

a 2 + b 2 = c 2. There are many proofs of this theorem. An elegant one only requires that we know that the area of a square of side L is L 2

a 2 + b 2 = c 2. There are many proofs of this theorem. An elegant one only requires that we know that the area of a square of side L is L 2 Pythgors Pythgors A right tringle, suh s shown in the figure elow, hs one 90 ngle. The long side of length is the hypotenuse. The short leg (or thetus) hs length, nd the long leg hs length. The theorem

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

8.2 Simplifying Radicals

8.2 Simplifying Radicals . Simplifig Rdicls I the lst sectio we sw tht sice. However, otice tht (-). So hs two differet squre roots. Becuse of this we eed to defie wht we cll the pricipl squre root so tht we c distiguish which

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

Roots, Radicals, and Complex Numbers

Roots, Radicals, and Complex Numbers 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

More information

Secondary Math 2 Honors. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers

Secondary Math 2 Honors. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Secodr Mth Hoors Uit Polomils, Epoets, Rdicls & Comple Numbers. Addig, Subtrctig, d Multiplig Polomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together. Moomils ol

More information

Chapter 6 Solving equations

Chapter 6 Solving equations Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign

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

Thank you for participating in Teach It First!

Thank you for participating in Teach It First! Thnk you for prtiipting in Teh It First! This Teh It First Kit ontins Common Core Coh, Mthemtis teher lesson followed y the orresponding student lesson. We re onfident tht using this lesson will help you

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

Functions A B C D E F G H I J K L. Contents:

Functions A B C D E F G H I J K L. Contents: Funtions Contents: A reltion is n set of points whih onnet two vriles. A funtion, sometimes lled mpping, is reltion in whih no two different ordered pirs hve the sme -oordinte or first omponent. Algeri

More information

Name Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE

Name Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE Name Date Block Know how to Algebra 1 Laws of Eponents/Polynomials Test STUDY GUIDE Evaluate epressions with eponents using the laws of eponents: o a m a n = a m+n : Add eponents when multiplying powers

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

ASYMPTOTES HORIZONTAL ASYMPTOTES VERTICAL ASYMPTOTES. An asymptote is a line which a function gets closer and closer to but never quite reaches.

ASYMPTOTES HORIZONTAL ASYMPTOTES VERTICAL ASYMPTOTES. An asymptote is a line which a function gets closer and closer to but never quite reaches. UNFAMILIAR FUNCTIONS (Chpter 19) 547 B ASYMPTOTES An smptote is line whih funtion gets loser n loser to but never quite rehes. In this ourse we onsier smptotes whih re horizontl or vertil. HORIZONTAL ASYMPTOTES

More information

ISTM206: Lecture 3 Class Notes

ISTM206: Lecture 3 Class Notes IST06: Leture 3 Clss otes ikhil Bo nd John Frik 9-9-05 Simple ethod. Outline Liner Progrmming so fr Stndrd Form Equlity Constrints Solutions, Etreme Points, nd Bses The Representtion Theorem Proof of the

More information

Right Triangle Trigonometry

Right Triangle Trigonometry CONDENSED LESSON 1.1 Right Tringle Trigonometr In this lesson ou will lern out the trigonometri rtios ssoited with right tringle use trigonometri rtios to find unknown side lengths in right tringle use

More information

excenters and excircles

excenters and excircles 21 onurrene IIi 2 lesson 21 exenters nd exirles In the first lesson on onurrene, we sw tht the isetors of the interior ngles of tringle onur t the inenter. If you did the exerise in the lst lesson deling

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

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

Are You Ready? Simplify Radical Expressions

Are You Ready? Simplify Radical Expressions SKILL Are You Read? Simplif Radical Epressions Teaching Skill Objective Simplif radical epressions. Review with students the definition of simplest form. Ask: Is written in simplest form? (No) Wh or wh

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

1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5.

1. 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 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

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

Homework 3 Solution Chapter 3.

Homework 3 Solution Chapter 3. Homework 3 Solution Chpter 3 2 Let Q e the group of rtionl numers under ddition nd let Q e the group of nonzero rtionl numers under multiplition In Q, list the elements in 1 2 In Q, list the elements in

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

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

10.5 Graphing Quadratic Functions

10.5 Graphing Quadratic Functions 0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions

More information

For the Final Exam, you will need to be able to:

For the Final Exam, you will need to be able to: Mth B Elementry Algebr Spring 0 Finl Em Study Guide The em is on Wednesdy, My 0 th from 7:00pm 9:0pm. You re lloed scientific clcultor nd " by 6" inde crd for notes. On your inde crd be sure to rite ny

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

Proving the Pythagorean Theorem

Proving the Pythagorean Theorem Proving the Pythgoren Theorem Proposition 47 of Book I of Eulid s Elements is the most fmous of ll Eulid s propositions. Disovered long efore Eulid, the Pythgoren Theorem is known y every high shool geometry

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

Three squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names.

Three squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names. 1- The Pythgoren Theorem MAIN IDEA Find length using the Pythgoren Theorem. New Voulry leg hypotenuse Pythgoren Theorem Mth Online glenoe.om Extr Exmples Personl Tutor Self-Chek Quiz Three squres with

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

to the area of the region bounded by the graph of the function y = f(x), the x-axis y = 0 and two vertical lines x = a and x = b.

to the area of the region bounded by the graph of the function y = f(x), the x-axis y = 0 and two vertical lines x = a and x = b. 5.9 Are in rectngulr coordintes If f() on the intervl [; ], then the definite integrl f()d equls to the re of the region ounded the grph of the function = f(), the -is = nd two verticl lines = nd =. =

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

The Quadratic Formula and the Discriminant

The Quadratic Formula and the Discriminant 9-9 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt

More information

Volumes by Cylindrical Shells: the Shell Method

Volumes 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 information

Napoleon and Pythagoras with Geometry Expressions

Napoleon and Pythagoras with Geometry Expressions Npoleon nd Pythgors with eometry xpressions NPOLON N PYTORS WIT OMTRY XPRSSIONS... 1 INTROUTION... xmple 1: Npoleon s Theorem... 3 xmple : n unexpeted tringle from Pythgors-like digrm... 5 xmple 3: Penequilterl

More information

Chess and Mathematics

Chess and Mathematics Chess nd Mthemtis in UK Seondry Shools Dr Neill Cooper Hed of Further Mthemtis t Wilson s Shool Mnger of Shool Chess for the English Chess Federtion Mths in UK Shools KS (up to 7 yers) Numers: 5 + 7; x

More information

CHAPTER 7: FACTORING POLYNOMIALS

CHAPTER 7: FACTORING POLYNOMIALS CHAPTER 7: FACTORING POLYNOMIALS FACTOR (noun) An of two or more quantities which form a product when multiplied together. 1 can be rewritten as 3*, where 3 and are FACTORS of 1. FACTOR (verb) - To factor

More information

Chapter15 SAMPLE. Simultaneous equations. Contents: A B C D. Graphical solution Solution by substitution Solution by elimination Problem solving

Chapter15 SAMPLE. Simultaneous equations. Contents: A B C D. Graphical solution Solution by substitution Solution by elimination Problem solving Chpter15 Simultneous equtions Contents: A B C D Grphil solution Solution y sustitution Solution y elimintion Prolem solving 308 SIMULTANEOUS EQUATIONS (Chpter 15) Opening prolem Ewen wnts to uy pie, ut

More information

II. SOLUTIONS TO HOMEWORK PROBLEMS Unit 1 Problem Solutions

II. SOLUTIONS TO HOMEWORK PROBLEMS Unit 1 Problem Solutions II. SOLUTIONS TO HOMEWORK PROLEMS Unit Prolem Solutions 757.25. (). () 23.7 6 757.25 6 47 r5 6 6 2 r5= 6 (4). r2 757.25 = 25.4 6 =. 2 2 5 4 6 23.7 6 7 r 6 r7 (2).72 6 ().52 6 (8).32. () 6 356.89 6 22 r4

More information

Addition and subtraction of rational expressions

Addition and subtraction of rational expressions Lecture 5. Addition nd subtrction of rtionl expressions Two rtionl expressions in generl hve different denomintors, therefore if you wnt to dd or subtrct them you need to equte the denomintors first. The

More information

Problem Set 2 Solutions

Problem Set 2 Solutions University of Cliforni, Berkeley Spring 2012 EE 42/100 Prof. A. Niknej Prolem Set 2 Solutions Plese note tht these re merely suggeste solutions. Mny of these prolems n e pprohe in ifferent wys. 1. In prolems

More information

Math Review for Algebra and Precalculus

Math 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 information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian 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 information

Digital Electronics Basics: Combinational Logic

Digital Electronics Basics: Combinational Logic Digitl Eletronis Bsis: for Bsi Eletronis http://ktse.eie.polyu.edu.hk/eie29 by Prof. Mihel Tse Jnury 25 Digitl versus nlog So fr, our disussion bout eletronis hs been predominntly nlog, whih is onerned

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

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

Right Triangle Trigonometry 8.7

Right Triangle Trigonometry 8.7 304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R1 8.7 Right Tringle Trigonometry R1 8.7 Right Tringle Trigonometry T E G T I N G S T R T E D The origins of trigonometry, from the Greek trigonon (ngle) nd

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

Reasoning to Solve Equations and Inequalities

Reasoning 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 information

15. Let f (x) = 3x Suppose rx 2 + sx + t = 0 where r 0. Then x = 24. Solve 5x 25 < 20 for x. 26. Let y = 7x

15. Let f (x) = 3x Suppose rx 2 + sx + t = 0 where r 0. Then x = 24. Solve 5x 25 < 20 for x. 26. Let y = 7x Pretest Review The pretest will onsist of 0 problems, eh of whih is similr to one of the following 49 problems If you n do problems like these 49 listed below, you will hve no problem with the pretest

More information

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S 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 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

Simplification Problems to Prepare for Calculus

Simplification Problems to Prepare for Calculus Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.

More information

Essential Question What are the Law of Sines and the Law of Cosines?

Essential Question What are the Law of Sines and the Law of Cosines? 9.7 TEXS ESSENTIL KNOWLEDGE ND SKILLS G.6.D Lw of Sines nd Lw of osines Essentil Question Wht re the Lw of Sines nd the Lw of osines? Disovering the Lw of Sines Work with prtner.. opy nd omplete the tle

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS 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 information

11. PYTHAGORAS THEOREM

11. PYTHAGORAS THEOREM 11. PYTHAGORAS THEOREM 11-1 Along the Nile 2 11-2 Proofs of Pythgors theorem 3 11-3 Finding sides nd ngles 5 11-4 Semiirles 7 11-5 Surds 8 11-6 Chlking hndll ourt 9 11-7 Pythgors prolems 10 11-8 Designing

More information

Angles and Triangles

Angles 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 information

SECTION 7-2 Law of Cosines

SECTION 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 information

COMPLEX FRACTIONS. section. Simplifying Complex Fractions

COMPLEX 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 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

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

MATHEMATICS I & II DIPLOMA COURSE IN ENGINEERING FIRST SEMESTER

MATHEMATICS I & II DIPLOMA COURSE IN ENGINEERING FIRST SEMESTER MATHEMATICS I & II DIPLOMA COURSE IN ENGINEERING FIRST SEMESTER A Plition nder Government of Tmilnd Distrition of Free Tetook Progrmme ( NOT FOR SALE ) Untohilit is sin Untohilit is rime Untohilit is inhmn

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

Ratio and Proportion

Ratio 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 information

4.0 5-Minute Review: Rational Functions

4.0 5-Minute Review: Rational Functions mth 130 dy 4: working with limits 1 40 5-Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two

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

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

Lesson 18.2: Right Triangle Trigonometry

Lesson 18.2: Right Triangle Trigonometry Lesson 8.: Right Tringle Trigonometry lthough Trigonometry is used to solve mny prolems, historilly it ws first pplied to prolems tht involve right tringle. This n e extended to non-right tringles (hpter

More information

Pythagoras theorem is one of the most popular theorems. Paper Folding And The Theorem of Pythagoras. Visual Connect in Teaching.

Pythagoras theorem is one of the most popular theorems. Paper Folding And The Theorem of Pythagoras. Visual Connect in Teaching. in the lssroom Visul Connet in Tehing Pper Folding And The Theorem of Pythgors Cn unfolding pper ot revel proof of Pythgors theorem? Does mking squre within squre e nything more thn n exerise in geometry

More information

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

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 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