SIMPLIFYING SQUARE ROOTS


 Thomasina Baker
 1 years ago
 Views:
Transcription
1 40 (88) Chapter 8 Powers and Roots 8. SIMPLIFYING SQUARE ROOTS In this section Using the Product Rule Rationalizing the Denominator Simplified Form of a Square Root In Section 8. you learned to simplify some radical expressions using the product rule. In this section you will learn three basic rules to follow for writing expressions involving square roots in simplest form. These rules can be extended to radicals with index greater than, but we will not do that in this text. Using the Product Rule We can use the product rule to simplify square roots of certain numbers. For example, 4 9 Factor 4 as Because 4 is not a perfect square, we cannot write 4 without the radical symbol. However, is considered a simpler expression that represents the exact value of 4. When simplifying square roots, we can factor the perfect squares out of the radical and replace them with their square roots. Look for the factors 4, 9,,,, 49, and so on. E X A M P L E calculator closeup You can use a calculator to see that and agree for the first 0 digits (out of infinitely many). Having the same first 0 digits does not make =. The product rule for radicals guarantees that they are equal. Simplifying radicals using the product rule Simplify. a) b) 0 c) a) Because 4, we can use the product rule to write 4. b) 0 c) Note that 4, 9, and are perfect squares and are factors of. In factoring out a perfect square, it is most efficient to use the largest perfect square: If we had factored out 9, we could still get the correct answer as follows: Rationalizing the Denominator Radicals such as,, and are irrational numbers. So a fraction such as has an irrational denominator. Because fractions with rational denominators are considered simpler than fractions with irrational denominators, we usually convert fractions with irrational denominators to equivalent ones with rational denominators. That is, we rationalize the denominator.
2 8. Simplifying Square Roots (89) 4 E X A M P L E Rationalizing denominators Simplify each expression by rationalizing its denominator. a) b) a) Because, we multiply numerator and denominator by : Multiply numerator and denominator by. b) Because, multiply the numerator and denominator by : Multiply numerator and denominator by. Simplified Form of a Square Root When we simplify any expression, we try to write a simpler expression that is equivalent to the original. However, one person s idea of simpler is sometimes different from another person s. For a square root the expression must satisfy three conditions to be in simplified form. These three conditions provide specific rules to follow for simplifying square roots. Simplified Form for Square Roots An expression involving a square root is in simplified form if it has. no perfectsquare factors inside the radical,. no fractions inside the radical, and. no radicals in the denominator. Because a decimal is a form of a fraction, a simplified square root should not contain any decimal numbers. Also, a simplified expression should use the fewest number of radicals possible. So we write rather than even though both and are both in simplified form. E X A M P L E Simplified form for square roots Write each radical expression in simplified form. a) 00 b) c) 0 a) We must remove the perfect square factor of 00 from inside the radical:
3 4 (80) Chapter 8 Powers and Roots calculator b) We first use the quotient rule to remove the fraction from inside the radical: closeup Using a calculator to check simplification problems will help you to understand the concepts. 0 Quotient rule for radicals c) The numerator and denominator have a common factor of : 0 Reduce. 0 Note that we could have simplified by first using the quotient rule to get 0 0 and then reducing 0. Another way to simplify 0 is to first multiply the numerator and denominator by. You should try these alternatives. Of course, the simplified form is by any method. In the next example we simplify some expressions involving variables. Remember that any exponential expression with an even exponent is a perfect square. E X A M P L E 4 Radicals containing variables Simplify each expression. All variables represent nonnegative real numbers. a) x b) 8a 9 c) 8a 4 b a) x x x The largest perfect square factor of x is x. x x xx For any nonnegative x, x x. b) 8a 9 4a 8 a The largest perfect square factor of 8a 9 is 4a 8. a 4 a 4a 8 a 4 c) 8a 4 b 9a 4 b b Factor out the perfect squares. a b b 9a 4 b a b
4 8. Simplifying Square Roots (8) 4 If square roots of variables appear in the denominator, then we rationalize the denominator. E X A M P L E helpful hint If you are going to compute the value of a radical expression with a calculator, it doesn t matter if the denominator is rational. However, rationalizing the denominator provides another opportunity to practice building up the denominator of a fraction and multiplying radicals. Radicals containing variables Simplify each expression. All variables represent positive real numbers. a) a b) a b c) a a a) a a a a a b) a b a b a b b b ab b Multiply numerator and denominator by a. a a a Quotient rule for radicals a c) a a a a a 4 a a a 4 a Factor out the perfect square. a a a a Factor the denominator. Divide out the common factor. CAUTION Do not attempt to reduce an expression like the one in Example (c): a a You cannot divide out common factors when one is inside a radical.
5 44 (8) Chapter 8 Powers and Roots WARMUPS True or false? Explain your answer.. 0 True. 8 9 False. True False. a aa for any positive value of a. True. a 9 a for any positive value of a. False. y y 8 y for any positive value of y. True 8. False 9. 4 False 0. 8 False 8. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. How do we simplify a radical with the product rule? We use the product rule to factor out a perfect square from inside a square root.. Which integers are perfect squares? The perfect squares are, 4, 9,,, and so on.. What does it mean to rationalize a denominator? To rationalize a denominator means to rewrite the expression so that the denominator is a rational number. 4. What is simplified form for a square root? A square root in simplified form has no perfect squares or fractions inside the radical and no radicals in the denominator.. How do you simplify a square root that contains a variable? To simplify a square root containing variables, use the same techniques as we use on square roots of numbers.. How can you tell if an exponential expression is a perfect square? Any even power of a variable is a perfect square. Assume that all variables in the exercises represent positive real numbers. Simplify each radical. See Example Simplify each expression by rationalizing the denominator. See Example Write each radical expression in simplified form. See Example
6 8. Simplifying Square Roots (8) 4 Simplify each expression. See Example a y 0 4. a 9 a 4 y a 4 a 4. t 4. 8a 48. 8w 9 t t a w 4 w 49. 0a 4 b 9 0. xy. xy a b 4 b xyy xyxy. 4xy. a b 8 c 4. xy z 9 4 x yxy ab 4 ca xy 4 z xy Simplify each expression. See Example.... x x a x x a x x a b y 0x 0b b y y x x. x y. w. 0 y x xy 0w xy y w x 4. 4 x y. 8 y xy y x xxy y s. 8 t s st t Simplify each expression.. 80x 8. 90y yx 9 4xx y 40 0 y 4 x yx 0x x y 0. 48xy.. x x 4xy y 4x x x yx p. 4. 0t. a b a b c 4 p p q t t a b c pq q 0t t a b 8 cac. n4 b n b c 4xy. 8m 8. n nb c x 9 y xy m n mn n 4 b c nb y n x 9 4 y m n Solve each problem. 8. Economic order quantity. The formula for economic order quantity E A I S was used in Exercise 8 of Section 8.. a) Express the righthand side in simplified form. E AIS I b) Find E when A, S $4, and I $80.. FIGURE FOR EXERCISE Landing speed. Aircraft design engineers determine the proper landing speed V (in ft/sec) by using the formula V 84L, CS where L is the gross weight of the aircraft in pounds, C is the coefficient of lift, and S is the wing surface area in square feet. a) Express the righthand side in simplified form. V 9 LCS CS b) Find V when L 800 pounds, C.8, and S 00 square feet..4 Use a calculator to evaluate each expression FIGURE FOR EXERCISE 84
Simplifying Numerical Square Root Expressions
10.1.1 Simplifying Numerical Square Root Expressions Definitions 1. The square of an integer is called a perfect square integer. Since 1 2 =1, 2 2 = 4, 3 2 = 9, 4 2 =16, etc..., the perfect square integers
More informationSimplification of Radical Expressions
8. Simplification of Radical Expressions 8. OBJECTIVES 1. Simplify a radical expression by using the product property. Simplify a radical expression by using the quotient property NOTE A precise set of
More informationSimplifying Radical Expressions
9.2 Simplifying Radical Expressions 9.2 OBJECTIVES. Simplify expressions involving numeric radicals 2. Simplify expressions involving algebraic radicals In Section 9., we introduced the radical notation.
More informationRADICALS 7.2. section. Radical Notation
7. Radicals (711) 97 b) The average radius of the orbit of Saturn is 9.0 AU. Use the accompanying graph to estimate the number of years it takes Saturn to make one orbit of the sun. a) 1. AU b) 7 years
More informationSection 9.1 Radical Expressions and Graphs
Chapter 9 Section 9.1 Radical Expressions and Graphs Objective: 1. Find square roots. 2. Decide whether a given root is rational, irrational, or not a real number. 3. Find cube, fourth, and other roots.
More informationDate: Section P.2: Exponents and Radicals. Properties of Exponents: Example #1: Simplify. a.) 3 4. b.) 2. c.) 3 4. d.) Example #2: Simplify. b.) a.
Properties of Exponents: Section P.2: Exponents and Radicals Date: Example #1: Simplify. a.) 3 4 b.) 2 c.) 34 d.) Example #2: Simplify. a.) b.) c.) d.) 1 Square Root: Principal n th Root: Example #3: Simplify.
More informationChapter 7: Radicals and Complex Numbers Lecture notes Math 1010
Section 7.1: Radicals and Rational Exponents Definition of nth root of a number Let a and b be real numbers and let n be an integer n 2. If a = b n, then b is an nth root of a. If n = 2, the root is called
More informationSquare Roots. Learning Objectives. PreActivity
Section 1. PreActivity Preparation Square Roots Our number system has two important sets of numbers: rational and irrational. The most common irrational numbers result from taking the square root of nonperfect
More informationChapter 7  Roots, Radicals, and Complex Numbers
Math 233  Spring 2009 Chapter 7  Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More informationNumbers, Operations, and Expressions. 1) Determine the classification(s) for each number below. List all that apply. 3
Numbers, Operations, and Expressions Review of Natural Numbers, Whole Numbers, Integers, and Rational Numbers 1) Determine the classification(s) for each number below. List all that apply. a) 11 b) 9.8
More informationExponents, Polynomials and Functions. Copyright Cengage Learning. All rights reserved.
Exponents, Polynomials and Functions 3 Copyright Cengage Learning. All rights reserved. 3.1 Rules for Exponents Copyright Cengage Learning. All rights reserved. Rules for Exponents The basic concept of
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 informationA.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents
Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify
More informationNegative Integer Exponents
7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions
More informationRadicals  Multiply and Divide Radicals
8. Radicals  Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals
More informationUSING THE PROPERTIES TO SIMPLIFY EXPRESSIONS
5 (1 5) Chapter 1 Real Numbers and Their Properties 1.8 USING THE PROPERTIES TO SIMPLIFY EXPRESSIONS In this section The properties of the real numbers can be helpful when we are doing computations. In
More informationRadicals and Rational Exponents
mes47759_ch0_6564 09/7/007 06:8 Page 65 pinnacle 0:MHIA08:mhmes:mesch0: CHAPTER 0 Radicals and Rational Exponents Algebra at Work: Forensics Forensic scientists use mathematics in many ways to help them
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 2 = x 5+2 = x 7 (x m ) n = x mn Example 2: (x 5 ) 2 = x 5 2 = x 10 (x m y n ) p = x mp y np Example
More informationP.2 Exponents and Radicals
71_0P0.qxp 1/7/06 9:0 AM Page 1 1 Chapter P Prerequisites P. Exponents and Radicals Integer Exponents Repeated multiplication can be written in exponential form. Repeated Multiplication Exponential Form
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationMULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.
1.4 Multiplication and (125) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with
More informationFINDING THE LEAST COMMON DENOMINATOR
0 (7 18) Chapter 7 Rational Expressions GETTING MORE INVOLVED 7. Discussion. Evaluate each expression. a) Onehalf of 1 b) Onethird of c) Onehalf of x d) Onehalf of x 7. Exploration. Let R 6 x x 0 x
More informationSTUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS
STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an
More informationUsing the Properties in Computation. a) 347 35 65 b) 3 435 c) 6 28 4 28
(1) Chapter 1 Real Numbers and Their Properties In this section 1.8 USING THE PROPERTIES TO SIMPLIFY EXPRESSIONS The properties of the real numbers can be helpful when we are doing computations. In this
More informationSimplifying SquareRoot Radicals Containing Perfect Square Factors
DETAILED SOLUTIONS AND CONCEPTS  OPERATIONS ON IRRATIONAL NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!
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 informationChapter 15 Radical Expressions and Equations Notes
Chapter 15 Radical Expressions and Equations Notes 15.1 Introduction to Radical Expressions The symbol is called the square root and is defined as follows: a = c only if c = a Sample Problem: Simplify
More information3 d A product of a number and a variable and x. 2c d Add and Subtract Polynomials
Math 50, Chapter 7 (Page 1 of 21) 7.1.1 Add and Subtract Polynomials Monomials A monomial is a number, a variable, or a product of numbers and variables. Examples of Monomials a. 7 A number b. 3 d A product
More information1.2 Exponents and Radicals. Copyright Cengage Learning. All rights reserved.
1.2 Exponents and Radicals Copyright Cengage Learning. All rights reserved. Objectives Integer Exponents Rules for Working with Exponents Scientific Notation Radicals Rational Exponents Rationalizing the
More informationDETAILED SOLUTIONS AND CONCEPTS INTRODUCTION TO IRRATIONAL AND IMAGINARY NUMBERS
DETAILED SOLUTIONS AND CONCEPTS INTRODUCTION TO IRRATIONAL AND IMAGINARY NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu.
More informationRational Exponents and Radicals
C H A P T E R 7 Rational Exponents and Radicals Wind chill temperature (F) for 5F air temperature 5 0 15 10 5 0 0.5 10 15 5 10 15 0 5 0 Wind velocity (mph) ust how cold is it in Fargo, North Dakota, in
More informationFactoring Polynomials
Factoring Polynomials Writing a polynomial as a product of polynomials of lower degree is called factoring. Factoring is an important procedure that is often used to simplify fractional expressions and
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More informationReal Numbers are used everyday to describe quantities such as age, weight, height, mpg, etc... Some common subsets of real numbers are:
P.1 Real Numbers and Their Properties Real Numbers are used everyday to describe quantities such as age, weight, height, mpg, etc... Some common subsets of real numbers are: Natural numbers N = {1, 2,
More informationMAT Make Your Own Study Guide Unit 3. Date Turned In
Name 14.1 Roots and Radicals Define perfect square. Date Turned In Example Show an example Show an example of a perfect square. Define square root. Show an example of a square root. What is the difference
More informationChapter 4  Decimals
Chapter 4  Decimals $34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value  1.23456789
More informationMath 2: Algebra 2, Geometry and Statistics Ms. SheppardBrick
Math : Algebra, Geometry and Statistics Ms. SheppardBrick 617.596.41 http://lps.lexingtonma.org/page/44 Math Chapter 1 Review Exponents and Radicals Exponent definitions and rules: For the expression,
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 informationMultiplying and Dividing Radical Expressions
Radicals DEAR Multiplying and Dividing Radical Expressions Learning Objective(s) Multiply and simplify radical expressions that contain a single term. Divide and simplify radical expressions that contain
More information1.1. Basic Concepts. Write sets using set notation. Write sets using set notation. Write sets using set notation. Write sets using set notation.
1.1 Basic Concepts Write sets using set notation. Objectives A set is a collection of objects called the elements or members of the set. 1 2 3 4 5 6 7 Write sets using set notation. Use number lines. Know
More information3.2 Equivalent Fractions: Simplifying and Building
3.2 Equivalent Fractions: Simplifying and Building Two fractions are said to be equivalent if they have the same value. Naturally, one approach we could use to determine if two fractions are equivalent
More informationRadicals. Stephen Perencevich
Radicals Stephen Perencevich Stephen Perencevich Georg Cantor Institute for Mathematical Studies Silver Spring, MD scpusa@gmail.com c 009 All rights reserved. Algebra II: Radicals 0 Introduction Perencevich
More informationThe notation above read as the nth root of the mth power of a, is a
Let s Reduce Radicals to Bare Bones! (Simplifying Radical Expressions) By Ana Marie R. Nobleza The notation above read as the nth root of the mth power of a, is a radical expression or simply radical.
More informationRadicals  Rational Exponents
8. Radicals  Rational Exponents Objective: Convert between radical notation and exponential notation and simplify expressions with rational exponents using the properties of exponents. When we simplify
More informationRational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have
8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents
More informationSect Exponents: Multiplying and Dividing Common Bases
40 Sect 5.1  Exponents: Multiplying and Dividing Common Bases Concept #1 Review of Exponential Notation In the exponential expression 4 5, 4 is called the base and 5 is called the exponent. This says
More informationGraphing Radicals STEM 7
Graphing Radicals STEM 7 Radical functions have the form: The most frequently used radical is the square root; since it is the most frequently used we assume the number 2 is used and the square root is
More informationDefinition of an nth Root
Radicals and Complex Numbers 7 7. Definition of an nth Root 7.2 Rational Exponents 7.3 Simplifying Radical Expressions 7.4 Addition and Subtraction of Radicals 7.5 Multiplication of Radicals 7.6 Rationalization
More informationFactors of 8 are 1 and 8 or 2 and 4. Let s substitute these into our factors and see which produce the middle term, 10x.
Quadratic equations A quadratic equation in x is an equation that can be written in the standard quadratic form ax + bx + c 0, a 0. Several methods can be used to solve quadratic equations. If the quadratic
More information6.2 FRACTIONAL EXPONENTS AND RADICAL EXPRESSIONS
Print this page 6.2 FRACTIONAL EXPONENTS AND RADICAL EXPRESSIONS A radical expression is an expression involving roots. For example, is the positive number whose square is a. Thus, since 3 2 = 9, and since
More informationMath 96Radicals #1 Simplify; Combinepage 1
Simplify; Combinepage 1 Part A Number Systems a. Whole Numbers = {0, 1, 2, 3,...} b. Integers = whole numbers and their opposites = {..., 3, 2, 1, 0, 1, 2, 3,...} c. Rational Numbers = quotient of integers
More informationMore generally, to approximate k, locate k between successive perfect squares. Then k must lie between their square roots. An example follows:
EXERCISE SET 10.1 DUE DATE: STUDENT: INSTRUCTOR: 10.1A Square Roots of Integers The square roots of a number are the values which, when squared, result in that number. If k is a square root of k, then
More informationSummer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2
Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level
More informationMath 002 Unit 5  Student Notes
Sections 7.1 Radicals and Radical Functions Math 002 Unit 5  Student Notes Objectives: Find square roots, cube roots, nth roots. Find where a is a real number. Look at the graphs of square root and cube
More informationTHE QUADRATIC FORMULA
66 (91) Chapter 9 Quadratic Equations and Quadratic Functions the members will sell 5000 00x tickets. So the total revenue for the tickets is given by R x (5000 00x). a) What is the revenue if the tickets
More informationRational Exponents. Given that extension, suppose that. Squaring both sides of the equation yields. a 2 (4 1/2 ) 2 a 2 4 (1/2)(2) a a 2 4 (2)
SECTION 0. Rational Exponents 0. OBJECTIVES. Define rational exponents. Simplify expressions with rational exponents. Estimate the value of an expression using a scientific calculator. Write expressions
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More information9.3 Solving Quadratic Equations by the Quadratic Formula
9.3 Solving Quadratic Equations by the Quadratic Formula OBJECTIVES 1 Identify the values of a, b, and c in a quadratic equation. Use the quadratic formula to solve quadratic equations. 3 Solve quadratic
More informationAlgebra Placement Test Review
Algebra Placement Test Review Recognizing the Relative Position between Real Numbers A. Which number is smaller, or 000? To really appreciate which number is smaller one must view both numbers plotted
More informationWhat you can do  (Goal Completion) Learning
What you can do  (Goal Completion) Learning ARITHMETIC READINESS Whole Numbers Order of operations: Problem type 1 Order of operations: Problem type 2 Factors Prime factorization Greatest common factor
More informationSimplifying Radical Expressions
Section 9 2A: Simplifying Radical Expressions Rational Numbers A Rational Number is any number that that expressed as a whole number a fraction a decimal that ends a decimal that repeats 3 2 1.2 1.333
More informationAdding Integers. Example 1 Evaluate.
Adding Integers Adding Integers 0 Example 1 Evaluate. Adding Integers Example 2 Evaluate. Adding Integers Example 3 Evaluate. Subtracting Integers Subtracting Integers Subtracting Integers Change the subtraction
More informationSimplifying Radical Expressions
In order to simplifying radical expression, it s important to understand a few essential properties. Product Property of Like Bases a a = a Multiplication of like bases is equal to the base raised to the
More informationChapter 1.1 Rational and Irrational Numbers
Chapter 1.1 Rational and Irrational Numbers A rational number is a number that can be written as a ratio or the quotient of two integers a and b written a/b where b 0. Integers, fractions and mixed numbers,
More informationRadicals  Multiply and Divide Radicals
8. Radicals  Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationeday Lessons Mathematics Grade 8 Student Name:
eday Lessons Mathematics Grade 8 Student Name: Common Core State Standards Expressions and Equations Work with radicals and integer exponents. 3. Use numbers expressed in the form of a single digit times
More informationPage 1 of Identify the degree of each term of the polynomial and the degree of the polynomial. The degree of the first term is.
1. Identify the degree of each term of the polynomial and the degree of the polynomial. x The degree of the first term is. The degree of the second term is. The degree of the third term is. The degree
More informationGeometry Summer Math Packet Review and Study Guide
V E R I T A S SAINT AGNES ACADEMY SAIN T DOMINIC SCHOOL Geometry Summer Math Packet Review and Study Guide This study guide is designed to aid students working on the Geometry Summer Math Packet. The purpose
More informationDifference of Squares and Perfect Square Trinomials
4.4 Difference of Squares and Perfect Square Trinomials 4.4 OBJECTIVES 1. Factor a binomial that is the difference of two squares 2. Factor a perfect square trinomial In Section 3.5, we introduced some
More informationNote that every natural number is an integer. There are integers (negative numbers) that are not natural numbers.
Real Numbers: Natural Numbers: N= {1, 2, 3, } Integers: Z= {0, 1, 1, 2, 2, 3, 3, } Note that every natural number is an integer. There are integers (negative numbers) that are not natural numbers. Rational
More informationCOGNITIVE TUTOR ALGEBRA
COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,
More information60 does not simplify. What s the flaw in
MTH 9 Radical Intervention Section 1 Simplifying Square Roots The square root of a number is not considered simplified if it contains a factor that is the perfect square of an integer (other than 1). For
More information7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radic. Expressions
7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radical Expressions Department of Mathematics Grossmont College November 5, 2012 Simplified Form for Radicals Learning Objectives: Write
More information2.1 Chapter 9 Concept 9.3: Zero, Negative,
2.. Chapter 9 Concept 9.: Zero, Negative, and Fractional Exponents Lesson) www.ck2.org 2. Chapter 9 Concept 9.: Zero, Negative, and Fractional Exponents Lesson) Simplify expressions with zero exponents.
More informationCHAPTER 8 ROOTS AND RADICALS
Chapter Eight Additional Exercises 09 CHAPTER 8 ROOTS AND RADICALS Section 8.1 Evaluating Roots Objective 1 Find square roots. Find all square roots of the number. 1. 81. 19.. 00... 1 8. 1 9. 9 10. 11
More information1.1 THE REAL NUMBERS. section. The Integers. The Rational Numbers
2 (1 2) Chapter 1 Real Numbers and Their Properties 1.1 THE REAL NUMBERS In this section In arithmetic we use only positive numbers and zero, but in algebra we use negative numbers also. The numbers that
More informationHow does the locations of numbers, variables, and operation signs in a mathematical expression affect the value of that expression?
How does the locations of numbers, variables, and operation signs in a mathematical expression affect the value of that expression? You can use powers to shorten how you present repeated multiplication.
More informationHFCC Math Lab Intermediate Algebra  17 DIVIDING RADICALS AND RATIONALIZING THE DENOMINATOR
HFCC Math Lab Intermediate Algebra  17 DIVIDING RADICALS AND RATIONALIZING THE DENOMINATOR Dividing Radicals: To divide radical expression we use Step 1: Simplify each radical Step 2: Apply the Quotient
More informationUNCORRECTED PAGE PROOFS
number and algebra TOPIC 6 Real numbers 6. Overview Why learn this? A knowledge of number is crucial if we are to understand the world around us. Over time, you have been building your knowledge of the
More informationThis is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0).
This is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES
ARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES 1. Squaring a number means using that number as a factor two times. 8 8(8) 64 (8) (8)(8) 64 Make sure students realize that x means (x ), not (x).
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationSimplifying Algebraic Expressions Involving Exponents
4.4 Simplifying Algebraic Expressions Involving Exponents GOAL Simplify algebraic expressions involving powers and radicals. LEARN ABOUT the Math The ratio of the surface area to the volume of microorganisms
More informationDefinition of Subtraction x  y = x + 1y2. Subtracting Real Numbers
Algebra Review Numbers FRACTIONS Addition and Subtraction i To add or subtract fractions with the same denominator, add or subtract the numerators and keep the same denominator ii To add or subtract fractions
More informationA square root function is a function whose rule contains a variable under a square root sign.
Chapter 111 SquareRoot Functions Part 1 A square root function is a function whose rule contains a variable under a square root sign. Example: Graph the squareroot function. Use a calculator to approximate
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationA. A Square Root: The square root of a number x is a number y such
CHAPTER 8 Section 8.1: Introduction to Square Roots and Radical Expressions In this chapter we will learn how to work with radical expressions such as square roots. A good background in this type of algebra
More informationRADICALS & RATIONAL EXPONENTS
c Gabriel Nagy RADICALS & RATIONAL EXPONENTS Facts about Power Equations Consider the power equation x N #, with N > integer and # any real number. Regarding the solvability of this equation, one has the
More information1.1 Solving a Linear Equation ax + b = 0
1.1 Solving a Linear Equation ax + b = 0 To solve an equation ax + b = 0 : (i) move b to the other side (subtract b from both sides) (ii) divide both sides by a Example: Solve x = 0 (i) x = 0 x = (ii)
More informationUnit 1 Review Part 1 3 combined Handout KEY.notebook. September 26, 2013
Math 10c Unit 1 Factors, Powers and Radicals Key Concepts 1.1 Determine the prime factors of a whole number. 650 3910 1.2 Explain why the numbers 0 and 1 have no prime factors. 0 and 1 have no prime factors
More informationCheck boxes of Edited Copy of Sp Topics (was 259 topics in pilot)
Check boxes of Edited Copy of 10022 Sp 11 258 Topics (was 259 topics in pilot) Beginning Algebra, 3rd Ed. [open all close all] Course Readiness and Additional Topics Appendix Course Readiness Multiplication
More informationThe Product Property of Square Roots states: For any real numbers a and b, where a 0 and b 0, ab = a b.
Chapter 9. Simplify Radical Expressions Any term under a radical sign is called a radical or a square root expression. The number or expression under the the radical sign is called the radicand. The radicand
More informationFACTORING OUT COMMON FACTORS
278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationChapter 1. Real Numbers Operations
www.ck1.org Chapter 1. Real Numbers Operations Review Answers 1 1. (a) 101 (b) 8 (c) 1 1 (d) 1 7 (e) xy z. (a) 10 (b) 14 (c) 5 66 (d) 1 (e) 7x 10 (f) y x (g) 5 (h) (i) 44 x. At 48 square feet per pint
More informationSOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The OddRoot Property
498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1
More information