Roots, Radicals, and Square Root Equations: Multiplication of Square Root Expressions *
|
|
|
- Audrey Harrison
- 5 years ago
- Views:
Transcription
1 OpenSta-CNX module: m Roots, Radicals, and Square Root Equations: Multiplication of Square Root Epressions * Wade Ellis Denn Burznski This work is produced b OpenSta-CNX and licensed under the Creative Commons Attribution License.0 Abstract This module is from Elementar Algebra b Denn Burznski and Wade Ellis, Jr. The distinction between the principal square root of the number and the secondar square root of the number is made b eplanation and b eample. The simplication of the radical epressions that both involve and do not involve fractions is shown in man detailed eamples; this is followed b an eplanation of how and wh radicals are eliminated from the denominator of a radical epression. Real-life applications of radical equations have been included, such as problems involving dail output, dail sales, electronic resonance frequenc, and kinetic energ. Objectives of this module: be able to use the product propert of square roots to multipl square roots. 1 Overview The Product Propert of Square Roots Multiplication Rule for Square Root Epressions 2 The Product Propert of Square Roots In our work with simplifing square root epressions, we noted that = Since this is an equation, we ma write it as = * Version 1.4: Jun 1, :4 pm
2 OpenSta-CNX module: m To multipl two square root epressions, we use the product propert of square roots. The Product Propert = = The product of the square roots is the square root of the product. In practice, it is usuall easier to simplif the square root epressions before actuall performing the multiplication. To see this, consider the following product: 4 We can multipl these square roots in either of two was: Eample 1 Simplif then multipl. Eample 2 Multipl then simplif. ( = 2 ) ( 2 4 ) = = 4 = 4 = 4 = 4 = Notice that in the second method, the epanded term (the third epression, 4) ma be dicult to factor into a perfect square and some other number. Multiplication Rule for Square Root Epressions The preceding eample suggests that the following rule for multipling two square root epressions. Rule for Multipling Square Root Epressions 1. Simplif each square root epression, if necessar. 2. Perform the multiplecation.. Simplif, if necessar. 4 Sample Set A Find each of the following products. Eample = = 1 = 9 2 = 2 Eample 4 2 = = = 2 4 = 2 2 = 4
3 OpenSta-CNX module: m2197 This product might be easier if we were to multipl rst and then simplif. 2 = 2 = 1 = 4 Eample 20 7 = 4 7 = 2 7 = 2 Eample a 27a = ( a a ) ( a 2 a ) = a 1a 2 = a a 1 = a 4 1 Eample 7 ( + 2) 7 1 = ( + 2) ( + 2) 1 = ( + 2) ( + 2) 1 Eample = ( + 2) ( + 2) ( 1) or = ( + 2) Eample 9 Eample 10
4 OpenSta-CNX module: m Practice Set A Find each of the following products. Eercise 1 (Solution on p. 9.) Eercise 2 (Solution on p. 9.) 2 2 Eercise (Solution on p. 9.) Eercise 4 (Solution on p. 9.) m n 20m 2 n Eercise (Solution on p. 9.) 9(k ) k 2 12k + Eercise (Solution on p. 9.) ( 2 + ) Eercise 7 (Solution on p. 9.) 2a ( a a ) Eercise (Solution on p. 9.) 2m n ( 2mn2 10n 7 ) Eercises Eercise 9 (Solution on p. 9.) 2 10 Eercise 10 1 Eercise 11 (Solution on p. 9.) 7 Eercise Eercise 1 (Solution on p. 9.) 2 27 Eercise Eercise 1 (Solution on p. 9.) Eercise Eercise 17 (Solution on p. 9.) Eercise Eercise 19 (Solution on p. 9.) 4 27
5 OpenSta-CNX module: m2197 Eercise Eercise 21 (Solution on p. 9.) m Eercise 22 7 a Eercise 2 (Solution on p. 9.) m Eercise h Eercise 2 (Solution on p. 9.) 20 a Eercise 2 4 Eercise 27 (Solution on p. 9.) 7 Eercise m Eercise 29 (Solution on p. 9.) a a Eercise 0 Eercise 1 (Solution on p. 9.) Eercise 2 h h Eercise (Solution on p. 9.) Eercise 4 Eercise (Solution on p. 9.) k k Eercise m m Eercise 7 (Solution on p. 9.) m 2 m Eercise a 2 a Eercise 9 (Solution on p. 9.) Eercise 40 Eercise 41 (Solution on p. 9.) 4
6 OpenSta-CNX module: m2197 Eercise 42 k k Eercise 4 (Solution on p. 10.) a a Eercise 44 7 Eercise 4 (Solution on p. 10.) 9 Eercise Eercise 47 (Solution on p. 10.) 4 Eercise 4 Eercise 49 (Solution on p. 10.) + 2 Eercise 0 a a + 1 Eercise 1 (Solution on p. 10.) + 2 Eercise 2 h + 1 h 1 Eercise (Solution on p. 10.) + 9 ( + 9) 2 Eercise 4 ( ) Eercise (Solution on p. 10.) a 2 1a Eercise 2m4 n 14m n Eercise 7 (Solution on p. 10.) 12(p q) (p q) Eercise 1a 2 (b + 4) 4 21a (b + 4) Eercise 9 12m n 4 r m r (Solution on p. 10.) Eercise 0 7(2k 1) 11 (k + 1) 14(2k 1) 10 Eercise 1 2 (Solution on p. 10.) Eercise Eercise (Solution on p. 10.) 2a 4 a 2a 7
7 OpenSta-CNX module: m Eercise 4 n n Eercise (Solution on p. 10.) 2 n 4 n Eercise a 2n+ a Eercise 7 (Solution on p. 10.) 2m n+1 10m n+ Eercise 7(a 2) 7 4a 9 Eercise ( 9 ) 2 + (Solution on p. 10.) Eercise 70 ( + 7 ) Eercise 71 (Solution on p. 10.) ( + 2 ) Eercise ( + ) Eercise 7 (Solution on p. 10.) ( a a ) Eercise 74 ( 2 4 ) Eercise 7 (Solution on p. 10.) ( + ) Eercise ( 2a 7 a a 11) Eercise 77 (Solution on p. 10.) ( 12m m7 m) Eercise 7 4 ( 7 ) 7 Eercises for Review Eercise 79 (Solution on p. 10.) () Factor a 4 4 2w 2. Eercise 0 () Find the slope of the line that passes through the points (, 4) and (, 4). Eercise 1 (Solution on p. 10.) () Perform the indicated operations:
8 OpenSta-CNX module: m2197 Eercise 2 () Simplif 4 2 z b removing the radical sign. Eercise (Solution on p. 10.) () Simplif 12 z.
9 OpenSta-CNX module: m Solutions to Eercises in this Module Solution to Eercise (p. 4) 0 Solution to Eercise (p. 4) Solution to Eercise (p. 4) ( + 4) ( + ) Solution to Eercise (p. 4) 4m n 10m Solution to Eercise (p. 4) (k ) 2 k Solution to Eercise (p. 4) + 1 Solution to Eercise (p. 4) a 10 4a 2 Solution to Eercise (p. 4) m n 2 n m 2 n m Solution to Eercise (p. 4) 2 Solution to Eercise (p. 4) 2 14 Solution to Eercise (p. 4) 12 Solution to Eercise (p. 4) Solution to Eercise (p. 4) Solution to Eercise (p. 4) Solution to Eercise (p. ) m Solution to Eercise (p. ) m Solution to Eercise (p. ) 2 a Solution to Eercise (p. ) Solution to Eercise (p. ) a Solution to Eercise (p. ) Solution to Eercise (p. ) Solution to Eercise (p. ) k Solution to Eercise (p. ) m m Solution to Eercise (p. ) 2
10 OpenSta-CNX module: m Solution to Eercise (p. ) 2 Solution to Eercise (p. ) a 4 Solution to Eercise (p. ) Solution to Eercise (p. ) Solution to Eercise (p. ) ( + 2) ( ) Solution to Eercise (p. ) ( + ) ( 2) Solution to Eercise (p. ) ( + 9) + 9 Solution to Eercise (p. ) a 2 a Solution to Eercise (p. ) (p q) 4 Solution to Eercise (p. ) 10m n 2 r 4 10mr Solution to Eercise (p. ) Solution to Eercise (p. ) 2a 7 Solution to Eercise (p. 7) n Solution to Eercise (p. 7) 2m 2n+2 Solution to Eercise (p. 7) 2 ( 2 + ) Solution to Eercise (p. 7) + Solution to Eercise (p. 7) 2 2a 2 a Solution to Eercise (p. 7) 2 ( + ) Solution to Eercise (p. 7) m 2 ( m 2 1 ) Solution to Eercise (p. 7) ( a w ) ( a 2 2 w ) Solution to Eercise (p. 7) 4(+) (+) 2 Solution to Eercise (p. ) 2 2 z 4
Ratios and Rates: Proportions *
OpenSta-CNX module: m3498 Ratios and Rates: Proportions * Wade Ellis Denny Burzynski This work is produced by OpenSta-CNX and licensed under the Creative Commons Attribution License 3.0 Abstract This module
Algebraic Expressions and Equations: Solving Equations of the Form x+a=b and x-a=b
OpenStax-CNX module: m35044 1 Algebraic Expressions and Equations: Solving Equations of the Form x+ab and x-ab Wade Ellis Denny Burzynski work is produced by OpenStax-CNX and licensed under the Creative
Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?
Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using
Algebraic Expressions and Equations: Classification of Expressions and Equations *
OpenStax-CNX module: m21848 1 Algebraic Expressions and Equations: Classification of Expressions and Equations * Wade Ellis Denny Burzynski This work is produced by OpenStax-CNX and licensed under the
Section 4.3: Quadratic Formula
Objective: Solve quadratic equations using the quadratic formula. In this section we will develop a formula to solve any quadratic equation ab c 0 where a b and c are real numbers and a 0. Solve for this
math FALL developmental mathematics sullivan 1e
TSIpractice eam review 1 131 180 plus 34 TSI questions for elementar and intermediate algebra m0300004301 aaa Name www.alvarezmathhelp.com math0300004301 FALL 01 100 interactmath developmental mathematics
Math 154 :: Elementary Algebra
Math :: Elementar Algebra Section. Section. Section. Section. Section. Section. Math :: Elementar Algebra Section. Eponents. When multipling like-bases, ou can add the eponents to simplif the epression..
MATH 108 REVIEW TOPIC 6 Radicals
Math 08 T6-Radicals Page MATH 08 REVIEW TOPIC 6 Radicals I. Computations with Radicals II. III. IV. Radicals Containing Variables Rationalizing Radicals and Rational Eponents V. Logarithms Answers to Eercises
Systems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
(1) Assignment # 1 Absolute Value. (2) Assignment # 2 Compound Absolute Values. (3) Assignment # 3 Exponents. (4) Assignment # 4 Simplifying Radicals
Alg_0 Packet # The beginning of our Quest () Assignment # Absolute Value () Assignment # Compound Absolute Values () Assignment # Eponents () Assignment # Simplifying Radicals (5) Assignment # 5 Fractional
Math 030 Review for Final Exam Revised Fall 2010 RH/ DM 1
Math 00 Review for Final Eam Revised Fall 010 RH/ DM 1 1. Solve the equations: (-1) (7) (-) (-1) () 1 1 1 1 f. 1 g. h. 1 11 i. 9. Solve the following equations for the given variable: 1 Solve for. D ab
Chapter 5: Systems of Equations and Inequalities. Section 5.4. Check Point Exercises
Chapter : Systems of Equations and Inequalities Section. Check Point Eercises. = y y = Solve the first equation for y. y = + Substitute the epression + for y in the second equation and solve for. ( + )
Ch 5 Alg 2 L2 Note Sheet Key Do Activity 1 on your Ch 5 Activity Sheet.
Ch Alg L Note Sheet Ke Do Activit 1 on our Ch Activit Sheet. Chapter : Quadratic Equations and Functions.1 Modeling Data With Quadratic Functions You had three forms for linear equations, ou will have
+ = + + = x = + = + = 36x
Ch 5 Alg L Homework Worksheets Computation Worksheet #1: You should be able to do these without a calculator! A) Addition (Subtraction = add the opposite of) B) Multiplication (Division = multipl b the
Math Departmental Exit Assessment Review (Student Version)
Math 03 - Departmental Eit Assessment Review (Student Version) Factor completel. (Factoring Review) ) 6-6 - 0 ) 3-343 3) 3 - + 8-40 4) - 36 ) + - 6 + 9 + 8 - + + Add or subtract as indicated. Simplif completel.
MI-2 Prob. Set #4 DUE: Tuesday, Feb 15, 2011 Spring 11. 4) Write a matrix equation and give result:
MI- Prob. Set #4 DUE: Tuesda, Feb 1, 011 Spring 11 1) Dr. Condie has sport coats, pairs of slacks, 4 shirts and 6 ties that he can choose to wear on an da. a) How man combinations of these four pieces
c) domain {x R, x 3}, range {y R}
Answers Chapter 1 Functions 1.1 Functions, Domain, and Range 1. a) Yes, no vertical line will pass through more than one point. b) No, an vertical line between = 6 and = 6 will pass through two points..
Northwest High School s Algebra 2/Honors Algebra 2
Northwest High School s Algebra /Honors Algebra Summer Review Packet 0 DUE Frida, September, 0 Student Name This packet has been designed to help ou review various mathematical topics that will be necessar
Section 5.1 Extra Practice
Name: Date: Section.1 Etra Practice BLM 1. Epress each radical as a simplified mied radical. 0 98, 0 6 y, 0, y 0. Epress each mied radical as an equivalent entire radical. 1 9, 0 y 7 y, 0, y 0. Order each
How can you write an equation of a line when you are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines
.7 Writing Equations in Point-Slope Form How can ou write an equation of a line when ou are given the slope and a point on the line? ACTIVITY: Writing Equations of Lines Work with a partner. Sketch the
Graphing Linear Equations and Inequalities: Proficiency Exam
Connexions module: m22015 1 Graphing Linear Equations and Inequalities: Proficiency Exam Denny Burzynski Wade Ellis This work is produced by The Connexions Project and licensed under the Creative Commons
Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs
Ch 5 Alg Note Sheet Ke Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Definition: Standard Form of a Quadratic Function The
Objectives To solve quadratic equations using the quadratic formula To find the number of solutions of a quadratic equation
9-6 The Quadratic Formula and the Discriminant Content Standards A.REI..a Use the method of completing the square to transform an quadratic equation in into an equation of the form ( p) 5 q... Derive the
Chapter 5: Systems of Equations
Chapter : Sstems of Equations Section.: Sstems in Two Variables... 0 Section. Eercises... 9 Section.: Sstems in Three Variables... Section. Eercises... Section.: Linear Inequalities... Section.: Eercises.
Increasing and decreasing intervals *
OpenStax-CNX module: m15474 1 Increasing and decreasing intervals * Sunil Kumar Singh This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 2.0 A function is
( 3x. Chapter Review. Review Key Vocabulary. Review Examples and Exercises 6.1 Properties of Square Roots (pp )
6 Chapter Review Review Ke Vocabular closed, p. 266 nth root, p. 278 eponential function, p. 286 eponential growth, p. 296 eponential growth function, p. 296 compound interest, p. 297 Vocabular Help eponential
Use the slope-intercept form to graph the equation. 8) 6x + y = 0
03 Review Solve the inequalit. Graph the solution set and write it in interval notation. 1) -2(4-9) < - + 2 Use the slope-intercept form to graph the equation. 8) 6 + = 0 Objective: (2.8) Solve Linear
Chapter 4 Page 1 of 16. Lecture Guide. Math College Algebra Chapter 4. to accompany. College Algebra by Julie Miller
Chapter 4 Page 1 of 16 Lecture Guide Math 105 - College Algebra Chapter 4 to accompan College Algebra b Julie Miller Corresponding Lecture Videos can be found at Prepared b Stephen Toner & Nichole DuBal
Math Lesson 2-2 Properties of Exponents
Math-00 Lesson - Properties of Eponents Properties of Eponents What is a power? Power: An epression formed b repeated multiplication of the base. Coefficient Base Eponent The eponent applies to the number
7.5 Solve Special Types of
75 Solve Special Tpes of Linear Sstems Goal p Identif the number of of a linear sstem Your Notes VOCABULARY Inconsistent sstem Consistent dependent sstem Eample A linear sstem with no Show that the linear
A. Simplifying Polynomial Expressions
A. Simplifing Polnomial Epressions I. Combining Like Terms - You can add or subtract terms that are considered "like", or terms that have the same variable(s) with the same eponent(s). E. 1: 5-7 + 10 +
10.1 Inverses of Simple Quadratic and Cubic Functions
COMMON CORE Locker LESSON 0. Inverses of Simple Quadratic and Cubic Functions Name Class Date 0. Inverses of Simple Quadratic and Cubic Functions Essential Question: What functions are the inverses of
Sample Problems For Grade 9 Mathematics. Grade. 1. If x 3
Sample roblems For 9 Mathematics DIRECTIONS: This section provides sample mathematics problems for the 9 test forms. These problems are based on material included in the New York Cit curriculum for 8.
Which of the following expressions are monomials?
9 1 Stud Guide Pages 382 387 Polnomials The epressions, 6, 5a 2, and 10cd 3 are eamples of monomials. A monomial is a number, a variable, or a product of numbers and variables. An eponents in a monomial
Preparing for A-Level Mathematics Summer 2017
Preparing for A-Level Mathematics Summer 017 INTRODUCTION TO A LEVEL MATHS Thank you for choosing to study Mathematics in the sith form. You will sit two modules in Pure Mathematics (C1 and C) as well
Limits of algebraic functions *
OpenStax-CNX module: m7542 Limits of algebraic functions * Sunil Kumar Singh This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 2.0 Algebraic expressions
Algebra/Pre-calc Review
Algebra/Pre-calc Review The following pages contain various algebra and pre-calculus topics that are used in the stud of calculus. These pages were designed so that students can refresh their knowledge
EXPONENT REVIEW!!! Concept Byte (Review): Properties of Exponents. Property of Exponents: Product of Powers. x m x n = x m + n
Algebra B: Chapter 6 Notes 1 EXPONENT REVIEW!!! Concept Byte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Property of Eponents: Product of Powers m n = m
Maths Department. A Level Induction Booklet
Maths Department A Level Induction Booklet CONTENTS Chapter 1 Removing brackets page Chapter Linear equations 4 Chapter 3 Simultaneous equations 8 Chapter 4 Factors 10 Chapter 5 Change the subject of the
A-LEVEL MATHS Bridging Work 2017
A-LEVEL MATHS Bridging Work 017 Name: Firstly, CONGRATULATIONS for choosing the best A-Level subject there is. A-Level Maths at Wales is not only interesting and enjoyable but is highly regarded by colleges,
Section Other Types of Equations
Section.5 - Other Types of Equations The numbers of solutions to a polynomial with n degree, where n is Natural Number, are n solutions. Solving a Polynomial Equation by factoring Eample Solve: = - = (
Further factorising, simplifying, completing the square and algebraic proof
Further factorising, simplifying, completing the square and algebraic proof 8 CHAPTER 8. Further factorising Quadratic epressions of the form b c were factorised in Section 8. by finding two numbers whose
3.6 Implicit Differentiation. explicit representation of a function rule. y = f(x) y = f (x) e.g. y = (x 2 + 1) 3 y = 3(x 2 + 1) 2 (2x)
Mathematics 0110a Summar Notes page 43 3.6 Implicit Differentiation eplicit representation of a function rule = f() = f () e.g. = ( + 1) 3 = 3( + 1) () implicit representation of a function rule f(, )
Elastic Collisions in One Dimension *
OpenStax-CNX module: m42163 1 Elastic Collisions in One Dimension * OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3.0 Abstract Describe an elastic
Math 111 Lecture Notes
A rational function is of the form R() = p() q() where p and q are polnomial functions. The zeros of a rational function are the values of for which p() = 0, as the function s value is zero where the value
Capacitors in Series and Parallel *
OpenStax-CNX module: m42336 Capacitors in Series and Parallel * OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 Abstract Derive expressions
Maths Department. A Level Induction Booklet
Maths Department A Level Induction Booklet One of the most important things if you are to succeed at A Level Maths is to ensure you understand all the algebra you met at GCSE. Working through the eamples
Algebra(2(Honors( Summer(Packet( ( ( Work(on(this(packet(during(the(summer.( ( ( There(will(be(an(in=class(quiz(on(the(material(the(
Algebra((Honors( Summer(Packet( ( ( Work(on(this(packet(during(the(summer.( ( ( There(will(be(an(in=class(quiz(on(the(material(the( first(week(of(school.( ( ( Please(visit(the(CHS(website(to(view(answers(to(
UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives
Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.
4 B. 4 D. 4 F. 3. How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?
3.1 Solving Quadratic Equations COMMON CORE Learning Standards HSA-SSE.A. HSA-REI.B.b HSF-IF.C.8a Essential Question Essential Question How can ou use the graph of a quadratic equation to determine the
Math-2. Lesson:1-2 Properties of Exponents
Math- Lesson:- Properties of Eponents Properties of Eponents What is a power? Power: An epression formed b repeated multiplication of the same factor. Coefficient Base Eponent The eponent applies to the
Math 0210 Common Final Review Questions (2 5 i)(2 5 i )
Math 0 Common Final Review Questions In problems 1 6, perform the indicated operations and simplif if necessar. 1. ( 8)(4) ( )(9) 4 7 4 6( ). 18 6 8. ( i) ( 1 4 i ) 4. (8 i ). ( 9 i)( 7 i) 6. ( i)( i )
10x y 7x y ELEMENTARY ALGEBRA. then x = GO ON TO THE NEXT PAGE. 1 16a is. 5. If 15 4y = 12, then y = 1. 3 x (A) (A) (D) 3x (B) (C) (D) (C) (D)
ELEMENTARY ALGEBRA This practice test measures our abilit to perform basic algebraic operations and to solve problems that involve elementar algebraic concepts. Directions: Solve each of the following
Summer Packet Geometry PAP
Summer Packet Geometry PAP IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Geometry with different strengths and needs. For this reason, students have options for completing
Course 15 Numbers and Their Properties
Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.
Algebra 1 Skills Needed for Success in Math
Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif
In everyday speech, a continuous. Limits and Continuity. Critical Thinking Exercises
062 Chapter Introduction to Calculus Critical Thinking Eercises Make Sense? In Eercises 74 77, determine whether each statement makes sense or does not make sense, and eplain our reasoning. 74. I evaluated
Section 7.3: Parabolas, from College Algebra: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons
Section 7.: Parabolas, from College Algebra: Corrected Edition b Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons Attribution-NonCommercial-ShareAlike.0 license. 0, Carl Stitz.
Vector (cross) product *
OpenStax-CNX module: m13603 1 Vector (cross) product * Sunil Kumar Singh This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 2.0 Abstract Vector multiplication
3.1 Graph Quadratic Functions
3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your
Glossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression
Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important
Summer Review For Students Entering Algebra 2
Summer Review For Students Entering Algebra Teachers and administrators at Tuscarora High School activel encourage parents and communit members to engage in children s learning. This Summer Review For
A11.1 Areas under curves
Applications 11.1 Areas under curves A11.1 Areas under curves Before ou start You should be able to: calculate the value of given the value of in algebraic equations of curves calculate the area of a trapezium.
BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH 05 Review Sheet
BRONX COMMUNITY COLLEGE of the Cit Universit of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH 05 Review Sheet Go to http://www.cun.edu/testing for more information on the CUNY Elementar Algebra
Module 2, Section 2 Solving Equations
Principles of Mathematics Section, Introduction 03 Introduction Module, Section Solving Equations In this section, you will learn to solve quadratic equations graphically, by factoring, and by applying
Chapter P Prerequisites: Fundamental Concepts of Algebra
Chapter P Prerequisites: Fundamental Concepts of Algebra Section P. Check Point Eercises. 8+ ( 8+ ( 8+ (0 8+ (00 8 + 00 08. Since 00 is 0 ears after 000, substitute 0 for. T 7 + + 7 7(0 + (0 + 7 77 If
5.3 Polynomials and Polynomial Functions
70 CHAPTER 5 Eponents, Polnomials, and Polnomial Functions 5. Polnomials and Polnomial Functions S Identif Term, Constant, Polnomial, Monomial, Binomial, Trinomial, and the Degree of a Term and of a Polnomial.
Chapter 18 Quadratic Function 2
Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are
Introduction to A-Level Maths (Bridging Unit)
Introduction to A-Level Maths (Bridging Unit) What is infinity + infinity? To infinity and beyond! SUMMER 017 Tuford Academy Faculty of Mathematics 1 INTRODUCTION TO A LEVEL MATHS AT TUXFORD ACADEMY Thank
11.1 Inverses of Simple Quadratic and Cubic Functions
Locker LESSON 11.1 Inverses of Simple Quadratic and Cubic Functions Teas Math Standards The student is epected to: A..B Graph and write the inverse of a function using notation such as f (). Also A..A,
Laurie s Notes. Overview of Section 3.5
Overview of Section.5 Introduction Sstems of linear equations were solved in Algebra using substitution, elimination, and graphing. These same techniques are applied to nonlinear sstems in this lesson.
Properties of Limits
33460_003qd //04 :3 PM Page 59 SECTION 3 Evaluating Limits Analticall 59 Section 3 Evaluating Limits Analticall Evaluate a it using properties of its Develop and use a strateg for finding its Evaluate
Section 4.1 Increasing and Decreasing Functions
Section.1 Increasing and Decreasing Functions The graph of the quadratic function f 1 is a parabola. If we imagine a particle moving along this parabola from left to right, we can see that, while the -coordinates
Chapter 9 Vocabulary Check
9 CHAPTER 9 Eponential and Logarithmic Functions Find the inverse function of each one-to-one function. See Section 9.. 67. f = + 68. f = - CONCEPT EXTENSIONS The formula = 0 e kt gives the population
Practice « «3. ` «- -2« 5. ` « « « $ «$ -7« $ 16.!2 $
Practice - Properties of Real Numbers Simplif.. - 4.«. - 6«. ` 7 4. «- -«6 `. ` 6. 0. -6«7. 4-8«8. -0.0«` Replace each $ with the smbol R, S,or to make the sentence true. 9.!6 $!0 0. $.. 0.06 $ 0.6. 4
Unit 2 Notes Packet on Quadratic Functions and Factoring
Name: Period: Unit Notes Packet on Quadratic Functions and Factoring Notes #: Graphing quadratic equations in standard form, verte form, and intercept form. A. Intro to Graphs of Quadratic Equations: a
Lesson 5: Negative Exponents and the Laws of Exponents
8 : Negative Eponents and the Laws of Eponents Student Outcomes Students know the definition of a number raised to a negative eponent. Students simplify and write equivalent epressions that contain negative
We have examined power functions like f (x) = x 2. Interchanging x
CHAPTER 5 Eponential and Logarithmic Functions We have eamined power functions like f =. Interchanging and ields a different function f =. This new function is radicall different from a power function
Add or subtract. Write the answer in lowest terms. 1) 15 13x 6. 13x A) 9 13x. 9 26x D) 9. B) 13x 9. 2) 2 r + 5 r 9 A) 18r 7 r(9 r) 7r 18 r(9 r)
Fall 2018 Preview Eam 4 Math 1 TTh This is intended onl as a resource to assist in our studing. This preview does not constitute acontract of inclusion and eclusion. MULTIPLE CHOICE. Choose the one alternative
10.4: WORKING WITH TAYLOR SERIES
04: WORKING WITH TAYLOR SERIES Contributed by OpenSta Mathematics at OpenSta CNX In the preceding section we defined Taylor series and showed how to find the Taylor series for several common functions
WCGS Mathematics Lower Sixth Bridging Work 2018
WCGS Mathematics Lower Sith Bridging Work 08 To be successful in Mathematics, students need to be confident in certain aspects of algebra, coordinate geometry and trigonometry before starting the course.
10.1 Inverses of Simple Quadratic and Cubic Functions
Name Class Date 10.1 Inverses of Simple Quadratic and Cubic Functions Essential Question: What functions are the inverses of quadratic functions and cubic functions, and how can ou find them? Resource
x Radical Sign: Radicand: the number beneath the radical sign
Sllabus Objective: 9.4 The student will solve quadratic equations using graphic and algebraic techniques to include the quadratic formula, square roots, factoring, completing the square, and graphing.
Mini-Lecture 7.1 Radicals and Radical Functions
Mini-Lecture 7. Radicals and Radical Functions Learning Objectives:. Find square roots.. Approimate roots.. Find cube roots.. Find n th roots.. Find n a n when a is an real number. 6. Graph square and
Name These exercises cover topics from Algebra I and Algebra II. Complete each question the best you can.
Name These eercises cover topics from Algebra I and Algebra II. Complete each question the best you can. Multiple Choice: Place through the letter of the correct answer. You may only use your calculator
Evaluate nth Roots and Use Rational Exponents. p Evaluate nth roots and study rational exponents. VOCABULARY. Index of a radical
. Georgia Performance Standard(s) MMA2a, MMA2b, MMAd Your Notes Evaluate nth Roots and Use Rational Eponents Goal VOCABULARY nth root of a p Evaluate nth roots and stud rational eponents. Inde of a radical
Linear and Nonlinear Systems of Equations. The Method of Substitution. Equation 1 Equation 2. Check (2, 1) in Equation 1 and Equation 2: 2x y 5?
3330_070.qd 96 /5/05 Chapter 7 7. 9:39 AM Page 96 Sstems of Equations and Inequalities Linear and Nonlinear Sstems of Equations What ou should learn Use the method of substitution to solve sstems of linear
Polynomial and Rational Functions
Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define
4.6 Model Direct Variation
4.6 Model Direct Variation Goal p Write and graph direct variation equations. Your Notes VOCABULARY Direct variation Constant of variation Eample Identif direct variation equations Tell whether the equation
math0320 FALL interactmath sections developmental mathematics sullivan 1e
Eam final eam review 180 plus 234 TSI questions for intermediate algebra m032000 013014 NEW Name www.alvarezmathhelp.com math0320 FALL 201 1400 interactmath sections developmental mathematics sullivan
APPENDIX D Rotation and the General Second-Degree Equation
APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the
LESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II
LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will
Unit 3 NOTES Honors Common Core Math 2 1. Day 1: Properties of Exponents
Unit NOTES Honors Common Core Math Da : Properties of Eponents Warm-Up: Before we begin toda s lesson, how much do ou remember about eponents? Use epanded form to write the rules for the eponents. OBJECTIVE
Name Class Date. Solving by Graphing and Algebraically
Name Class Date 16-4 Nonlinear Sstems Going Deeper Essential question: How can ou solve a sstem of equations when one equation is linear and the other is quadratic? To estimate the solution to a sstem
14.1 Systems of Linear Equations in Two Variables
86 Chapter 1 Sstems of Equations and Matrices 1.1 Sstems of Linear Equations in Two Variables Use the method of substitution to solve sstems of equations in two variables. Use the method of elimination
Mini Lecture 9.1 Finding Roots
Mini Lecture 9. Finding Roots. Find square roots.. Evaluate models containing square roots.. Use a calculator to find decimal approimations for irrational square roots. 4. Find higher roots. Evaluat. a.
Alperton Community School. Preparation for. A Level Mathematics. This induction booklet is for students who wish to start AS Level Maths in Year 12.
Alperton Community School Preparation for A Level Mathematics This induction booklet is for students who wish to start AS Level Maths in Year 1. You are epected to know these topics before your first maths
Fair Game Review. Chapter 9. Find the square root(s) ± Find the side length of the square. 7. Simplify Simplify 63.
Name Date Chapter 9 Find the square root(s). Fair Game Review... 9. ±. Find the side length of the square.. s. s s Area = 9 ft s Area = 0. m 7. Simplif 0. 8. Simplif. 9. Simplif 08. 0. Simplif 88. Copright
Fitting Integrands to Basic Rules
6_8.qd // : PM Page 8 8 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8. Basic Integration Rules Review proceres for fitting an integrand to one of the basic integration
Algebra 1 Skills Needed to be Successful in Algebra 2
Algebra 1 Skills Needed to be Successful in Algebra A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed