7.5 Rationalizing Denominators and Numerators of Radical Expressions
|
|
- Adela Price
- 6 years ago
- Views:
Transcription
1 440 CHAPTER Rational Exponents, Radicals, and Complex Numbers 86. Find the area and perimeter of the trapezoid. (Hint: The area of a trapezoid is the product of half the height 6 meters and the sum of the bases 6 and meters.) m 6 m m 6 m 8. Add: +. b. Multiply: #. c. Describe the differences in parts (a) and (b). 88. Add: + b. Multiply: # c. Describe the differences in parts (a) and (b). 89. Multiply: Multiply: Explain how simplifying x + x is similar to simplifying x + x. 9. Explain how multiplying x - x + is similar to multiplying x - x +.. Rationalizing Denominators and Numerators of Radical Expressions S Rationalize Denominators. Rationalize Denominators Having Two Terms. Rationalize Numerators. Rationalizing Denominators of Radical Expressions Often in mathematics, it is helpful to write a radical expression such as either without a radical in the denominator or without a radical in the numerator. The process of writing this expression as an equivalent expression but without a radical in the denominator is called rationalizing the denominator. To rationalize the denominator of, we use the fundamental principle of fractions and multiply the numerator and the denominator by. Recall that this is the same as multiplying by, which simplifies to. # # In this section, we continue to assume that variables represent positive real numbers. EXAMPLE Solution Rationalize the denominator of each expression. b. 6 9x c. A To rationalize the denominator, we multiply the numerator and denominator by a factor that makes the radicand in the denominator a perfect square. # # The denominator is now rationalized. b. First, we simplify the radicals and then rationalize the denominator. 6 9x 4 x 8 x To rationalize the denominator, multiply the numerator and denominator by x. Then 8 x 8 # x x # x 8x x
2 Section. Rationalizing Denominators and Numerators of Radical Expressions 44 c. A. Now we rationalize the denominator. Since is a cube root, we want to multiply by a value that will make the radicand a perfect cube. If we multiply by, we get # # Rationalize the denominator of each expression. b. 4x Multiply the numerator and denominator by and then simplify. c. A 9 CONCEPT CHECK Determine by which number both the numerator and denominator can be multiplied to rationalize the denominator of the radical expression. b. 4 8 EXAMPLE Solution x A y x y Rationalize the denominator of A x y. x # y y # y xy y Rationalize the denominator of A z y. Use the quotient rule. No radical may be simplified further. Multiply numerator and denominator by y so that the radicand in the denominator is a perfect square. Use the product rule in the numerator and denominator. Remember that y # y y. EXAMPLE Rationalize the denominator of Solution First, simplify each radical if possible. 4 x 4 x 4 8y 4 8y 4 # 4 y 4 x y 4 y 4 x # 4 y y 4 y # 4 y 4 xy y 4 y 4 4 xy y 4 x 4 8y. Use the product rule in the denominator. Write 4 8y 4 as y. Multiply numerator and denominator by 4 y so that the radicand in the denominator is a perfect fourth power. Use the product rule in the numerator and denominator. In the denominator, 4 y 4 y and y # y y. Answer to Concept Check: or 49 b. 4 Rationalize the denominator of z x 4.
3 44 CHAPTER Rational Exponents, Radicals, and Complex Numbers Rationalizing Denominators Having Two Terms Remember the product of the sum and difference of two terms? a + ba - b a - b a Q These two expressions are called conjugates of each other. To rationalize a numerator or denominator that is a sum or difference of two terms, we use conjugates. To see how and why this works, let s rationalize the denominator of the expression. To do so, we multiply both the numerator and the denominator - by +, the conjugate of the denominator -, and see what happens Multiply the sum and difference of two terms: a + ba - b a - b. - + or Notice in the denominator that the product of - and its conjugate, +, is -. In general, the product of an expression and its conjugate will contain no radical terms. This is why, when rationalizing a denominator or a numerator containing two terms, we multiply by its conjugate. Examples of conjugates are a - b and a + b x + y and x - y EXAMPLE Solution Rationalize each denominator. b c. m x + m Multiply the numerator and denominator by the conjugate of the denominator, , or - 4 It is often useful to leave a numerator in factored form to help determine whether the expression can be simplified.
4 Section. Rationalizing Denominators and Numerators of Radical Expressions 44 b. Multiply the numerator and denominator by the conjugate of c. Multiply by the conjugate of x + m to eliminate the radicals from the denominator. m x + m mx - m x + mx - m 6mx - m 9x - m 6mx - m x - m 4 Rationalize the denominator. + b. + - c. x x + y Rationalizing Numerators As mentioned earlier, it is also often helpful to write an expression such as as an equivalent expression without a radical in the numerator. This process is called rationalizing the numerator. To rationalize the numerator of, we multiply the numerator and the denominator by. # # EXAMPLE Rationalize the numerator of 4. Solution First we simplify # Next we rationalize the numerator by multiplying the numerator and the denominator by. # # # Rationalize the numerator of 80.
5 444 CHAPTER Rational Exponents, Radicals, and Complex Numbers EXAMPLE 6 Rationalize the numerator of x y. Solution The numerator and the denominator of this expression are already simplified. To rationalize the numerator, x, we multiply the numerator and denominator by a factor that will make the radicand a perfect cube. If we multiply x by 4x, we get 8x x. x y x # 4x y # 4x 8x 0xy x 0xy 6 Rationalize the numerator of b a. EXAMPLE Rationalize the numerator of x +. Solution We multiply the numerator and the denominator by the conjugate of the numerator, x +. x + x + x - x - x - x - x - 4 x - Rationalize the numerator of x - 4 Multiply by x -, the conjugate of x +. a + ba - b a - b. Vocabulary, Readiness & Video Check Use the choices below to fill in each blank. Not all choices will be used. rationalizing the numerator conjugate rationalizing the denominator. The of a + b is a - b.. The process of writing an equivalent expression, but without a radical in the denominator is, called.. The process of writing an equivalent expression, but without a radical in the numerator, is called. 4. To rationalize the denominator of, we multiply by. Martin-Gay Interactive Videos See Video. Watch the section lecture video and answer the following questions.. From Examples, what is the goal of rationalizing a denominator? 6. From Example 4, why will multiplying a denominator by its conjugate always rationalize the denominator?. From Example, is the process of rationalizing a numerator any different from rationalizing a denominator?
6 Section. Rationalizing Denominators and Numerators of Radical Expressions 44. Exercise Set Rationalize each denominator. See Examples through A 4 A x 4 8x x. 9 a x A y A x A 0 z y 4. A A y 6 9 a y x x A A9x 0. - a A b A0 y B 4 x x 4y 4 4 A9. Am 6 n a 9y. 4. 8a 9 b 4 4y 9 Write the conjugate of each expression.. + x 6. + y. - a b x y Rationalize each denominator. See Example x y a + a - b x x + y a - a + b a x - y Rationalize each numerator. See Examples and A 8 A 4x y 4x A x A8 x 0 8x 4 y 6 B z A A x 6 4x z 4 A y A 9y A 8x y B z Rationalize each numerator. See Example x + x - + x + x x x + y x - y
7 446 CHAPTER Rational Exponents, Radicals, and Complex Numbers REVIEW AND PREVIEW Solve each equation. See Sections. and x - x x - 4 x - 8. x - 6x y + y x - 8x x x CONCEPT EXTENSIONS 89. The formula of the radius r of a sphere with surface area A is A r A 4p Rationalize the denominator of the radical expression in this formul 90. The formula for the radius r of a cone with height centimeters and volume V is V r A p Rationalize the numerator of the radical expression in this formul r cm r y 9. Given rationalize the denominator by following x, parts (a) and (b). Multiply the numerator and denominator by x. b. Multiply the numerator and denominator by x. c. What can you conclude from parts (a) and (b)? 9. Given y, rationalize the denominator by following parts 4 (a) and (b). Multiply the numerator and denominator by 6. b. Multiply the numerator and denominator by. c. What can you conclude from parts (a) and (b)? Determine the smallest number both the numerator and denominator should be multiplied by to rationalize the denominator of the radical expression. See the Concept Check in this section When rationalizing the denominator of, explain why both the numerator and the denominator must be multiplied by. 96. When rationalizing the numerator of, explain why both the numerator and the denominator must be multiplied by. 9. Explain why rationalizing the denominator does not change the value of the original expression. 98. Explain why rationalizing the numerator does not change the value of the original expression. Integrated Review RADICALS AND RATIONAL EXPONENTS Sections.. Throughout this review, assume that all variables represent positive real numbers. Find each root A x 6. y y 0. -y b Use radical notation to write each expression. Simplify if possible / 0. y / /. x + / Use the properties of exponents to simplify each expression. Write with positive exponents.. y -/6 # y /6 4. x/ 4 x /6. x/4 x /4 x -/ / # 4 / Use rational exponents to simplify each radical.. 8x 6 8. a 9 b 6
8 Section.6 Radical Equations and Problem Solving 44 Use rational exponents to write each as a single radical expression x # x 0. # Simplify x y 0. 4x b 0 Multiply or divide. Then simplify if possible.. # x 6. 8x # 8x. 98y a 9 b y 4 ab Perform each indicated operation y 4 - y 6y x - x + 4. x + - Rationalize each denominator.. A Rationalize each numerator. 8. A 6. x. 9. A 9y x - x.6 Radical Equations and Problem Solving S Solve Equations That Contain Radical Expressions. Use the Pythagorean Theorem to Model Problems. Solving Equations That Contain Radical Expressions In this section, we present techniques to solve equations containing radical expressions such as x - 9 We use the power rule to help us solve these radical equations. Power Rule If both sides of an equation are raised to the same power, all solutions of the original equation are among the solutions of the new equation. This property does not say that raising both sides of an equation to a power yields an equivalent equation. A solution of the new equation may or may not be a solution of the original equation. For example, -, but -. Thus, each solution of the new equation must be checked to make sure it is a solution of the original equation. Recall that a proposed solution that is not a solution of the original equation is called an extraneous solution. EXAMPLE Solve: x - 9. Solution We use the power rule to square both sides of the equation to eliminate the radical. x - 9 x - 9 x - 8 x 84 x 4 Now we check the solution in the original equation. (Continued on next page)
7.4 Adding, Subtracting, and Multiplying Radical Expressions. OBJECTIVES 1 Add or Subtract Radical Expressions. 2 Multiply Radical Expressions.
CHAPTER 7 Rational Exponents, Radicals, and Complex Numbers Find and correct the error. See the Concept Check in this section. 11. 116. 6 6 = 6 A6 = 1 = 1 16 = 16 A = Simplify. See a Concept Check in this
More information2.2 Radical Expressions I
2.2 Radical Expressions I Learning objectives Use the product and quotient properties of radicals to simplify radicals. Add and subtract radical expressions. Solve real-world problems using square root
More informationSection 3.7: Solving Radical Equations
Objective: Solve equations with radicals and check for extraneous solutions. In this section, we solve equations that have roots in the problem. As you might expect, to clear a root we can raise both sides
More information7.2 Rational Exponents
Section 7.2 Rational Exponents 49 7.2 Rational Exponents S Understand the Meaning of a /n. 2 Understand the Meaning of a m/n. 3 Understand the Meaning of a -m/n. 4 Use Rules for Exponents to Simplify Expressions
More informationLESSON 9.1 ROOTS AND RADICALS
LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical
More informationNever leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!
1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a
More informationChapter 4: Radicals and Complex Numbers
Section 4.1: A Review of the Properties of Exponents #1-42: Simplify the expression. 1) x 2 x 3 2) z 4 z 2 3) a 3 a 4) b 2 b 5) 2 3 2 2 6) 3 2 3 7) x 2 x 3 x 8) y 4 y 2 y 9) 10) 11) 12) 13) 14) 15) 16)
More informationWorking with Square Roots. Return to Table of Contents
Working with Square Roots Return to Table of Contents 36 Square Roots Recall... * Teacher Notes 37 Square Roots All of these numbers can be written with a square. Since the square is the inverse of the
More informationSection 1.1 Task List
Summer 017 Math 143 Section 1.1 7 Section 1.1 Task List Section 1.1 Linear Equations Work through Section 1.1 TTK Work through Objective 1 then do problems #1-3 Work through Objective then do problems
More informationIntermediate Algebra
Intermediate Algebra George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 102 George Voutsadakis (LSSU) Intermediate Algebra August 2013 1 / 40 Outline 1 Radicals
More informationSOLUTIONS FOR PROBLEMS 1-30
. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).
More information5.1 Multiplying and Dividing Radical Expressions
Practice 5. Multiplying and Dividing Radical Expressions Multiply, if possible. Then simplify. To start, identify the index of each radical.. 4 # 6. 5 # 8. 6 # 4 9 index of both radicals is 4 # 6 Simplify.
More informationLesson 2. When the exponent is a positive integer, exponential notation is a concise way of writing the product of repeated factors.
Review of Exponential Notation: Lesson 2 - read to the power of, where is the base and is the exponent - if no exponent is denoted, it is understood to be a power of 1 - if no coefficient is denoted, it
More informationUse properties of exponents. Use the properties of rational exponents to simplify the expression. 12 d.
EXAMPLE 1 Use properties of exponents Use the properties of rational exponents to simplify the expression. a. 7 1/ 7 1/2 7 (1/ + 1/2) 7 / b. (6 1/2 1/ ) 2 (6 1/2 ) 2 ( 1/ ) 2 6( 1/2 2 ) ( 1/ 2 ) 6 1 2/
More informationEquations and Inequalities. College Algebra
Equations and Inequalities College Algebra Radical Equations Radical Equations: are equations that contain variables in the radicand How to Solve a Radical Equation: 1. Isolate the radical expression on
More informationRadical Expressions and Graphs 8.1 Find roots of numbers. squaring square Objectives root cube roots fourth roots
8. Radical Expressions and Graphs Objectives Find roots of numbers. Find roots of numbers. The opposite (or inverse) of squaring a number is taking its square root. Find principal roots. Graph functions
More information10.1. Square Roots and Square- Root Functions 2/20/2018. Exponents and Radicals. Radical Expressions and Functions
10 Exponents and Radicals 10.1 Radical Expressions and Functions 10.2 Rational Numbers as Exponents 10.3 Multiplying Radical Expressions 10.4 Dividing Radical Expressions 10.5 Expressions Containing Several
More informationNatural Numbers Positive Integers. Rational Numbers
Chapter A - - Real Numbers Types of Real Numbers, 2,, 4, Name(s) for the set Natural Numbers Positive Integers Symbol(s) for the set, -, - 2, - Negative integers 0,, 2,, 4, Non- negative integers, -, -
More informationEquations, Inequalities, and Problem Solving
CHAPTER Equations, Inequalities, and Problem Solving. Linear Equations in One Variable. An Introduction to Problem Solving. Formulas and Problem Solving.4 Linear Inequalities and Problem Solving Integrated
More informationIntroductory Algebra Chapter 9 Review
Introductory Algebra Chapter 9 Review Objective [9.1a] Find the principal square roots and their opposites of the whole numbers from 0 2 to 2 2. The principal square root of a number n, denoted n,is the
More informationHerndon High School Geometry Honors Summer Assignment
Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in
More informationGeometry Summer Assignment
2018-2019 Geometry Summer Assignment You must show all work to earn full credit. This assignment will be due Friday, August 24, 2018. It will be worth 50 points. All of these skills are necessary to be
More informationSummary for a n = b b number of real roots when n is even number of real roots when n is odd
Day 15 7.1 Roots and Radical Expressions Warm Up Write each number as a square of a number. For example: 25 = 5 2. 1. 64 2. 0.09 3. Write each expression as a square of an expression. For example: 4. x
More informationMini 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.
More informationLesson 28: A Focus on Square Roots
now Lesson 28: A Focus on Square Roots Student Outcomes Students solve simple radical equations and understand the possibility of extraneous solutions. They understand that care must be taken with the
More informationUNIT 4: RATIONAL AND RADICAL EXPRESSIONS. 4.1 Product Rule. Objective. Vocabulary. o Scientific Notation. o Base
UNIT 4: RATIONAL AND RADICAL EXPRESSIONS M1 5.8, M2 10.1-4, M3 5.4-5, 6.5,8 4.1 Product Rule Objective I will be able to multiply powers when they have the same base, including simplifying algebraic expressions
More informationChapter 4: Radicals and Complex Numbers
Chapter : Radicals and Complex Numbers Section.1: A Review of the Properties of Exponents #1-: Simplify the expression. 1) x x ) z z ) a a ) b b ) 6) 7) x x x 8) y y y 9) x x y 10) y 8 b 11) b 7 y 1) y
More informationAlgebra 1 Unit 6 Notes
Algebra 1 Unit 6 Notes Name: Day Date Assignment (Due the next class meeting) Monday Tuesday Wednesday Thursday Friday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday
More informationSection 10.1 Radical Expressions and Functions. f1-152 = = = 236 = 6. 2x 2-14x + 49 = 21x = ƒ x - 7 ƒ
78 CHAPTER 0 Radicals, Radical Functions, and Rational Exponents Chapter 0 Summary Section 0. Radical Expressions and Functions If b a, then b is a square root of a. The principal square root of a, designated
More informationEquations and Inequalities
Equations and Inequalities 2 Figure 1 CHAPTER OUTLINE 2.1 The Rectangular Coordinate Systems and Graphs 2.2 Linear Equations in One Variable 2.3 Models and Applications 2.4 Complex Numbers 2.5 Quadratic
More informationJUST THE MATHS UNIT NUMBER 1.3. ALGEBRA 3 (Indices and radicals (or surds)) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1 ALGEBRA (Indices and radicals (or surds)) by AJHobson 11 Indices 12 Radicals (or Surds) 1 Exercises 14 Answers to exercises UNIT 1 - ALGEBRA - INDICES AND RADICALS (or Surds)
More information4.1 Estimating Roots Name: Date: Goal: to explore decimal representations of different roots of numbers. Main Ideas:
4.1 Estimating Roots Name: Goal: to explore decimal representations of different roots of numbers Finding a square root Finding a cube root Multiplication Estimating Main Ideas: Definitions: Radical: an
More informationWe will work with two important rules for radicals. We will write them for square roots but they work for any root (cube root, fourth root, etc.).
College algebra We will review simplifying radicals, exponents and their rules, multiplying polynomials, factoring polynomials, greatest common denominators, and solving rational equations. Pre-requisite
More informationNote: In this section, the "undoing" or "reversing" of the squaring process will be introduced. What are the square roots of 16?
Section 8.1 Video Guide Introduction to Square Roots Objectives: 1. Evaluate Square Roots 2. Determine Whether a Square Root is Rational, Irrational, or Not a Real Number 3. Find Square Roots of Variable
More informationSUMMER REVIEW PACKET. Name:
Wylie East HIGH SCHOOL SUMMER REVIEW PACKET For students entering Regular PRECALCULUS Name: Welcome to Pre-Calculus. The following packet needs to be finished and ready to be turned the first week of the
More informationChapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18,
More informationMath 2 Variable Manipulation Part 2 Powers & Roots PROPERTIES OF EXPONENTS:
Math 2 Variable Manipulation Part 2 Powers & Roots PROPERTIES OF EXPONENTS: 1 EXPONENT REVIEW PROBLEMS: 2 1. 2x + x x + x + 5 =? 2. (x 2 + x) (x + 2) =?. The expression 8x (7x 6 x 5 ) is equivalent to?.
More informationExtending the Number System
Analytical Geometry Extending the Number System Extending the Number System Remember how you learned numbers? You probably started counting objects in your house as a toddler. You learned to count to ten
More informationWarm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2
8-8 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Simplify each expression. Assume all variables are positive. 1. 2. 3. 4. Write each expression in radical form. 5. 6. Objective Solve radical equations
More informationEquations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero
Solving Other Types of Equations Rational Equations Examine each denominator to find values that would cause the denominator to equal zero Multiply each term by the LCD or If two terms cross-multiply Solve,
More informationIrrational Numbers Study Guide
Square Roots and Cube Roots Positive Square Roots A positive number whose square is equal to a positive number b is denoted by the symbol b. The symbol b is automatically denotes a positive number. The
More information2-6 Nonlinear Inequalities
31. Find the domain of each expression. For the expression to be defined, x 2 3x 40 0. Let f (x) = x 2 3x 40. A factored form of f (x) is f (x) = (x 8)(x + 5). f (x) has real zeros at x = 8 and x = 5.
More informationAn equation is a statement that states that two expressions are equal. For example:
Section 0.1: Linear Equations Solving linear equation in one variable: An equation is a statement that states that two expressions are equal. For example: (1) 513 (2) 16 (3) 4252 (4) 64153 To solve the
More informationSolve a radical equation
EXAMPLE 1 Solve a radical equation Solve 3 = 3. x+7 3 x+7 = 3 Write original equation. 3 ( x+7 ) 3 = 3 3 x+7 = 7 x = 0 x = 10 Cube each side to eliminate the radical. Subtract 7 from each side. Divide
More informationSection 7.1 Rational Functions and Simplifying Rational Expressions
Beginning & Intermediate Algebra, 6 th ed., Elayn Martin-Gay Sec. 7.1 Section 7.1 Rational Functions and Simplifying Rational Expressions Complete the outline as you view Video Lecture 7.1. Pause the video
More informationRational Numbers. a) 5 is a rational number TRUE FALSE. is a rational number TRUE FALSE
Fry Texas A&M University!! Math 150!! Chapter 1!! Fall 2014! 1 Chapter 1A - - Real Numbers Types of Real Numbers Name(s) for the set 1, 2,, 4, Natural Numbers Positive Integers Symbol(s) for the set, -,
More informationChapter 3: Factors, Roots, and Powers
Chapter 3: Factors, Roots, and Powers Section 3.1 Chapter 3: Factors, Roots, and Powers Section 3.1: Factors and Multiples of Whole Numbers Terminology: Prime Numbers: Any natural number that has exactly
More informationRadicals: To simplify means that 1) no radicand has a perfect square factor and 2) there is no radical in the denominator (rationalize).
Summer Review Packet for Students Entering Prealculus Radicals: To simplify means that 1) no radicand has a perfect square factor and ) there is no radical in the denominator (rationalize). Recall the
More informationMATH 190 KHAN ACADEMY VIDEOS
MATH 10 KHAN ACADEMY VIDEOS MATTHEW AUTH 11 Order of operations 1 The Real Numbers (11) Example 11 Worked example: Order of operations (PEMDAS) 7 2 + (7 + 3 (5 2)) 4 2 12 Rational + Irrational Example
More informationP.1 Prerequisite skills Basic Algebra Skills
P.1 Prerequisite skills Basic Algebra Skills Topics: Evaluate an algebraic expression for given values of variables Combine like terms/simplify algebraic expressions Solve equations for a specified variable
More informationChapter 8 RADICAL EXPRESSIONS AND EQUATIONS
Name: Instructor: Date: Section: Chapter 8 RADICAL EXPRESSIONS AND EQUATIONS 8.1 Introduction to Radical Expressions Learning Objectives a Find the principal square roots and their opposites of the whole
More informationAlgebra I. Exponents and Polynomials. Name
Algebra I Exponents and Polynomials Name 1 2 UNIT SELF-TEST QUESTIONS The Unit Organizer #6 2 LAST UNIT /Experience NAME 4 BIGGER PICTURE DATE Operations with Numbers and Variables 1 CURRENT CURRENT UNIT
More informationPowers, Roots and Radicals. (11) Page #23 47 Column, #51, 54, #57 73 Column, #77, 80
Algebra 2/Trig Unit Notes Packet Name: Period: # Powers, Roots and Radicals () Homework Packet (2) Homework Packet () Homework Packet () Page 277 # 0 () Page 277 278 #7 6 Odd (6) Page 277 278 #8 60 Even
More informationMultiplying a Monomial times a Monomial. To multiply a monomial term times a monomial term with radicals you use the following rule A B C D = A C B D
Section 7 4A: Multiplying Radical Expressions Multiplying a Monomial times a Monomial To multiply a monomial term times a monomial term with radicals you use the following rule A B C D = A C B D In other
More informationREVIEW, pages
REVIEW, pages 156 161 2.1 1. For each pair of numbers, write two expressions to represent the distance between the numbers on a number line, then determine this distance. a) 1 8 and 1 4 b) 7.5 and -.75
More informationEXPONENT 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
More informationALGEBRA 2 Summer Review Assignments Graphing
ALGEBRA 2 Summer Review Assignments Graphing To be prepared for algebra two, and all subsequent math courses, you need to be able to accurately and efficiently find the slope of any line, be able to write
More informationExponents, Radicals, and Scientific Notation
General Exponent Rules: Exponents, Radicals, and Scientific Notation x m x n = x m+n Example 1: x 5 x = x 5+ = x 7 (x m ) n = x mn Example : (x 5 ) = x 5 = x 10 (x m y n ) p = x mp y np Example : (x) =
More informationChapter 5: Exponents and Polynomials
Chapter 5: Exponents and Polynomials 5.1 Multiplication with Exponents and Scientific Notation 5.2 Division with Exponents 5.3 Operations with Monomials 5.4 Addition and Subtraction of Polynomials 5.5
More informationChapter 1.6. Perform Operations with Complex Numbers
Chapter 1.6 Perform Operations with Complex Numbers EXAMPLE Warm-Up 1 Exercises Solve a quadratic equation Solve 2x 2 + 11 = 37. 2x 2 + 11 = 37 2x 2 = 48 Write original equation. Subtract 11 from each
More informationALGEBRA I CURRICULUM OUTLINE
ALGEBRA I CURRICULUM OUTLINE 2013-2014 OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines
More informationDefinitions Term Description Examples Mixed radical the product of a monomial and a radical
Chapter 5 Radical Expressions and Equations 5.1 Working With Radicals KEY IDEAS Definitions Term Description Examples Mixed radical the product of a monomial and a radical index radical sign -8 45 coefficient
More informationSECTION Types of Real Numbers The natural numbers, positive integers, or counting numbers, are
SECTION.-.3. Types of Real Numbers The natural numbers, positive integers, or counting numbers, are The negative integers are N = {, 2, 3,...}. {..., 4, 3, 2, } The integers are the positive integers,
More informationA field trips costs $800 for the charter bus plus $10 per student for x students. The cost per student is represented by: 10x x
LEARNING STRATEGIES: Activate Prior Knowledge, Shared Reading, Think/Pair/Share, Note Taking, Group Presentation, Interactive Word Wall A field trips costs $800 for the charter bus plus $10 per student
More information27 = 3 Example: 1 = 1
Radicals: Definition: A number r is a square root of another number a if r = a. is a square root of 9 since = 9 is also a square root of 9, since ) = 9 Notice that each positive number a has two square
More informationRational Expressions
CHAPTER 6 Rational Epressions 6. Rational Functions and Multiplying and Dividing Rational Epressions 6. Adding and Subtracting Rational Epressions 6.3 Simplifying Comple Fractions 6. Dividing Polynomials:
More informationLesson 9: Radicals and Conjugates
Lesson 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
More informationChapter 3. September 11, ax + b = 0.
Chapter 3 September 11, 2017 3.1 Solving equations Solving Linear Equations: These are equations that can be written as ax + b = 0. Move all the variables to one side of the equation and all the constants
More information2.5 Compound Inequalities
Section.5 Compound Inequalities 89.5 Compound Inequalities S 1 Find the Intersection of Two Sets. Solve Compound Inequalities Containing and. Find the Union of Two Sets. 4 Solve Compound Inequalities Containing
More informationMath 1302 Notes 2. How many solutions? What type of solution in the real number system? What kind of equation is it?
Math 1302 Notes 2 We know that x 2 + 4 = 0 has How many solutions? What type of solution in the real number system? What kind of equation is it? What happens if we enlarge our current system? Remember
More informationNOTES: EXPONENT RULES
NOTES: EXPONENT RULES DAY 2 Topic Definition/Rule Example(s) Multiplication (add exponents) x a x b = x a+b x 4 x 8 x 5 y 2 x 2 y Power to a Power (multiply exponents) x a ( ) b = x ab ( x ) 7 ( x ) 2
More information7.6 Radical Equations and Problem Solving
Section 7.6 Radical Equations and Problem Solving 447 Use rational eponents to write each as a single radical epression. 9. 2 4 # 2 20. 25 # 2 3 2 Simplify. 2. 240 22. 2 4 6 7 y 0 23. 2 3 54 4 24. 2 5-64b
More informationRadical Expressions, Equations, and Functions
Radical Expressions, Equations, and Functions 0 Real-World Application An observation deck near the top of the Sears Tower in Chicago is 353 ft high. How far can a tourist see to the horizon from this
More informationAccuplacer Review Workshop. Elementary Algebra Part II. Week Three. Includes internet links to instructional videos for additional resources:
Accuplacer Review Workshop Elementary Algebra Part II Week Three Includes internet links to instructional videos for additional resources: http://www.mathispower4u.com (Arithmetic Video Library) http://www.purplemath.com
More informationB.3 Solving Equations Algebraically and Graphically
B.3 Solving Equations Algebraically and Graphically 1 Equations and Solutions of Equations An equation in x is a statement that two algebraic expressions are equal. To solve an equation in x means to find
More information4.3 Mixed and Entire Radicals
4.3 Mixed and Entire Radicals Index Review of Radicals Radical 3 64 Radicand When the index of the radical is not shown then it is understood to be an index of 2. 64 = 2 64 MULTIPLICATION PROPERTY of RADICALS
More information2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY
2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY The following are topics that you will use in Geometry and should be retained throughout the summer. Please use this practice to review the topics you
More informationPRECALCULUS GUIDED NOTES FOR REVIEW ONLY
PRECALCULUS GUIDED NOTES Contents 1 Number Systems and Equations of One Variable 1 1.1 Real Numbers and Algebraic Expressions................ 1 1.1.a The Real Number System.................... 1 1.1.b
More informationUnit 9 Study Sheet Rational Expressions and Types of Equations
Algebraic Fractions: Unit 9 Study Sheet Rational Expressions and Types of Equations Simplifying Algebraic Fractions: To simplify an algebraic fraction means to reduce it to lowest terms. This is done by
More informationPrerequisites. Introduction CHAPTER OUTLINE
Prerequisites 1 Figure 1 Credit: Andreas Kambanls CHAPTER OUTLINE 1.1 Real Numbers: Algebra Essentials 1.2 Exponents and Scientific Notation 1. Radicals and Rational Expressions 1. Polynomials 1. Factoring
More informationB. Complex number have a Real part and an Imaginary part. 1. written as a + bi some Examples: 2+3i; 7+0i; 0+5i
Section 11.8 Complex Numbers I. The Complex Number system A. The number i = -1 1. 9 and 24 B. Complex number have a Real part and an Imaginary part II. Powers of i 1. written as a + bi some Examples: 2+3i;
More information1Add and subtract 2Multiply radical
Then You simplified radical expressions. (Lesson 10-2) Now 1Add and subtract radical expressions. 2Multiply radical expressions. Operations with Radical Expressions Why? Conchita is going to run in her
More informationAnswers (Lesson 11-1)
Answers (Lesson -) Lesson - - Study Guide and Intervention Product Property of Square Roots The Product Property of Square Roots and prime factorization can be used to simplify expressions involving irrational
More informationRadical Expressions and Functions What is a square root of 25? How many square roots does 25 have? Do the following square roots exist?
Topic 4 1 Radical Epressions and Functions What is a square root of 25? How many square roots does 25 have? Do the following square roots eist? 4 4 Definition: X is a square root of a if X² = a. 0 Symbolically,
More informationMAC 1105 Lecture Outlines for Ms. Blackwelder s lecture classes
MAC 1105 Lecture Outlines for Ms. Blackwelder s lecture classes These notes are prepared using software that is designed for typing mathematics; it produces a pdf output. Alternative format is not available.
More informationSections 7.2, 7.3, 4.1
Sections 7., 7.3, 4.1 Section 7. Multiplying, Dividing and Simplifying Radicals This section will discuss the rules for multiplying, dividing and simplifying radicals. Product Rule for multiplying radicals
More informationMath ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying
Math 1050 2 ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying General Tips for Studying: 1. Review this guide, class notes, the
More informationSimplifying Radicals. multiplication and division properties of square roots. Property Multiplication Property of Square Roots
10-2 Simplifying Radicals Content Standard Prepares for A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Objective To simplify
More informationAlgebra II Vocabulary Alphabetical Listing. Absolute Maximum: The highest point over the entire domain of a function or relation.
Algebra II Vocabulary Alphabetical Listing Absolute Maximum: The highest point over the entire domain of a function or relation. Absolute Minimum: The lowest point over the entire domain of a function
More informationDear Future Pre-Calculus Students,
Dear Future Pre-Calculus Students, Congratulations on your academic achievements thus far. You have proven your academic worth in Algebra II (CC), but the challenges are not over yet! Not to worry; this
More informationSection A.7 and A.10
Section A.7 and A.10 nth Roots,,, & Math 1051 - Precalculus I Roots, Exponents, Section A.7 and A.10 A.10 nth Roots & A.7 Solve: 3 5 2x 4 < 7 Roots, Exponents, Section A.7 and A.10 A.10 nth Roots & A.7
More informationExponents, Polynomials, and Polynomial Functions
CHAPTER Exponents, Polynomials, and Polynomial Functions. Exponents and Scientific Notation. More Work with Exponents and Scientific Notation. Polynomials and Polynomial Functions. Multiplying Polynomials.
More informationA Correlation of. Pearson. Mathematical Ideas. to the. TSI Topics
A Correlation of Pearson 2016 to the A Correlation of 2016 Table of Contents Module M1. Linear Equations, Inequalities, and Systems... 1 Module M2. Algebraic Expressions and Equations (Other Than Linear)...
More information1.4 Properties of Real Numbers and Algebraic Expressions
0 CHAPTER Real Numbers and Algebraic Expressions.4 Properties of Real Numbers and Algebraic Expressions S Use Operation and Order Symbols to Write Mathematical Sentences. 2 Identify Identity Numbers and
More informationRational Exponents Connection: Relating Radicals and Rational Exponents. Understanding Real Numbers and Their Properties
Name Class 6- Date Rational Exponents Connection: Relating Radicals and Rational Exponents Essential question: What are rational and irrational numbers and how are radicals related to rational exponents?
More informationSolving Multi-Step Equations
1. Clear parentheses using the distributive property. 2. Combine like terms within each side of the equal sign. Solving Multi-Step Equations 3. Add/subtract terms to both sides of the equation to get the
More informationMath 005A Prerequisite Material Answer Key
Math 005A Prerequisite Material Answer Key 1. a) P = 4s (definition of perimeter and square) b) P = l + w (definition of perimeter and rectangle) c) P = a + b + c (definition of perimeter and triangle)
More informationLet me just tell you how, then I ll use an example to make it more clear.
CHAPTER 15 Equations with Radicals Sec. 1 Simplifying Radicals From grade school, you can probably remember how to take square roots of numbers like 25, 64, and the 100. The number on the inside of the
More informationTo solve a radical equation, you must take both sides of an equation to a power.
Topic 5 1 Radical Equations A radical equation is an equation with at least one radical expression. There are four types we will cover: x 35 3 4x x 1x 7 3 3 3 x 5 x 1 To solve a radical equation, you must
More informationConcept Category 4. Quadratic Equations
Concept Category 4 Quadratic Equations 1 Solving Quadratic Equations by the Square Root Property Square Root Property We previously have used factoring to solve quadratic equations. This chapter will introduce
More information