This is a square root. The number under the radical is 9. (An asterisk * means multiply.)


 Delphia Miller
 1 years ago
 Views:
Transcription
1 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 denominators. Find domain of a radical function. Evaluate expressions involving rational exponents. Solve a radical equation. Evaluate square roots or cube roots. Example: Evaluate 9 This is a square root. The number under the radical is 9. (An asterisk * means multiply.) 9 Find a number such that the number squared is 9. This number is because 9 Example: Evaluate 8 This is a cube root. The number under the radical is 8. 8 Find a number such that the number cubed is 8. This number is because 8.
2 Page of Example: Evaluate 400 This is a square root. The number under the radical is does not have a real number solution. Find a number such that the number squared is 400. Note that and (not 400). Any real number squared is a positive number or zero. Example: Evaluate 400 This is a square root. The number under the radical is Find a number such that the number squared is 400. ( ) Thus 0 0 Don t forget the negative sign in front of. Example: Evaluate 7 This is a cube root. The number under the radical is 7. ( ) 7 Find a number such that the number cubed will give you 7. The number is because ( ) ( ) ( ) ( ) 7.
3 Page of Simplify radical expressions and Rationalize denominators Usually to simplify means to rewrite the expression in such a way that it has as few radicals as possible, and that the expression under each radical does not contain perfect powers. Some rules for radicals are illustrated below. a. or ( ) b. *0 0 c d. 5 * 7 5*7 5 e *5 0 Also, ( 7 ) 7 7 and 5 (+ 5) 6 * Keep in mind that: does not equal 5 + 7, and 5 5 does not equal. 7 7 Example: Simplify 8. The only thing which can be simplified is 8 under the square root. Can it be factored in such a way that one of the multiples is a perfect square? 8 4*7 *7 * 7 7 Factor 8 so one factor is a perfect square. The radical of a product equals the product of the radicals. Simplify the perfect square.
4 Page 4 of Example: Simplify 7x. Assume x 0. The expression under the square root has a perfect square. 7x 7 x 7 x The radical of a product equals the product of the radicals. Simplify the perfect square. Example: Simplify 5 k. Assume k 0. The radical is a square root. The expression under the square root is not a perfect square. 5 4 k k k 4 k k k k. Factor so there are perfect squares. The radical of a product equals the product of the radicals. Simplify the perfect square. k remains under the radical since the exponent on k is which is smaller than the index () of the radical Example: Simplify 7 8 k. Assume k 0 and q 0. 0 q The radical is a square root. The expression under the square root is not a perfect square kq 0 k q 5 64 k k q k k q 5 8 k k q kq k Factor so there are perfect squares. The radical of a product equals the product of the radicals. Simplify the perfect squares. The radical is simplified because there are no perfect squares left under the radical and the remaining variables and numbers have exponents of one.
5 Page 5 of Example: Simplify 7k 5. Assume k 0. This is a cube root. Each factor is a perfect cube. and k ( k ) 7k 5 5 The radical of a product equals the product of k 7 5 k 5 ( k ) the radicals. Write each factor as a perfect cube. Simplify the cube roots. Example: Simplify x y. Assume x 0 and y 0. The radical is a cube root. The expression under the cube root is not a perfect cube. 64x y x y The radical of a product is the product of the 4 4 x xy xy x x x x y x y y y y y radicals. Factor so there are perfect cubes. Simplify the perfect cubes. Rearrange the factors. Generally the radical factor is written last The radical is simplified because there are no perfect cubes left in the radical.
6 Page 6 of Example: Simplify t 8t. Assume t 0. The expression is the product of two square roots. t 8t t 8t The product of two radicals equals the radical of their product. 4t Multiply. 4 t The radical of the product equals the product of the radicals. 6 t Factor so there are perfect squares. Simplify the square roots and rearrange. t 6 Example: Simplify y 4 y + 5. Assume y. The expression is the product of two square roots. y 4 y 4 The product of two radicals equals the radical of the product. y + 5 y + 5 ( y ) 4 ( y + 5) ( y ) y + 5 y y + 5 Multiply. Factor the perfect square in the numerator. Distribute in the denominator. Simplify the perfect square.
7 Page 7 of Example: Simplify 6. 6 The expression is the quotient of two cube roots The quotient of two radicals equals the radical of their quotient. Divide. Simplify the perfect cube. 7 Example: Simplify The expression is the sum of two radicals. The radicals are like radicals. Both contain (4 + 8) 5 Add the like radicals. 5 Example: Simplify The expression contains the sum of four radical terms. The like radicals can be added Rearrange so like radicals are together. ( 4 + 9) 5 + ( + ) Combine the like radicals
8 Page 8 of Example: Simplify The expression contains the sum of two square roots Simplify the radicals to determine if there are like radicals Combine the like radicals. 7 Example: Simplify x 8 + x 8. Assume x 8. The expression contains the sum of two terms containing radicals. x 8 + x 8 4 x 8 The radicals in both terms are like radicals ( x 8 ), so the terms can be added. Example: Simplify x + + 9x + 8. Assume x. The expression contains the sum of two terms. x + + 9x + 8 x + + 9( x + ) Simplify the radicals. x + + x + + x + 4 x + ( x + ) Combine the like radicals.
9 Page 9 of Example: Rationalize the denominator. 5 Rationalize the denominator means to find an equivalent fraction whose denominator does not contain a radical ( 5) Multiply the numerator and the denominator by 5. (Note: You can t just square the numerator and the denominator. It will change the value of the fraction) Simplify the denominator.
10 Page 0 of Example: Rationalize denominator The denominator of this expression is irrational because it includes an irrational number ( ). Rationalize denominator means to find an equivalent expression, but with a rational denominator. 5 (5 )(5 ) 5+ (5+ )(5 ) ( ) There is a binomial in the denominator. Multiply the numerator and denominator by ( 5 ). (The conjugate of the denominator.) Find the product of the numerator and denominator. Combine like radicals. Note: Choosing ( 5 ) uses the formula ( a b)( a b) a b and b. Squaring The fraction would change the value of the fraction. +. In this example, a 5
11 Page of Find the domain of a function Domain The domain is a list or set of all possible inputs that yield a real number output. There are three operations we can t do with real numbers in algebra. Each of these restrict the domain. Can t divide by zero. Can t take the square root (or any evenindex radical) of a negative number. Can t take the logarithm of zero or a negative number. Two common notations to write the domain are setbuilder and interval notation.. Setbuilder notation: Sets are typically written in braces { }. The notation is { independent variable some property or restriction about independent variable } where the vertical line is read such that. Example: All real numbers, x, less than. { xx< } Example: All real numbers, n 4 n< 6 { } n, greater than or equal to 4 and less than 6.. Interval notation: Parenthesis indicate the starting or ending value is not included and a square bracket indicates the starting or ending value is included. Within the parentheses or square bracket, we indicate the smallest value of x followed by a comma and then the largest value of x. The examples above are shown using interval notation. Example: All real numbers, x, less than., ( ) Example: All real numbers, 4,6 [ ) n, greater than or equal to 4 and less than 6.
12 Page of Find the domain of a radical function Example: Find the domain of y x. The function contains a square root. The expression under the square root, x, must be greater or equal to zero. x 0 x x 6 In interval notation the answer is [ 6, ) Isolate the term with a variable. Divide both sides by. ( is positive, so don t change the inequality sign) Example: Find the domain of y 5 t. The function contains a square root. The expression under the square root, 5 t, must be greater or equal to zero. 5 t 0 t 5 5 t In interval notation the answer is 5,. Isolate the term with a variable. Divide both sides by. (Remember when you multiply or divide an inequality by a negative number, the inequality sign changes direction.)
13 Page of Example: Find the domain of y x + 4. The function contains cube root. The expression under the cube root can be any real number. x + 4 x + 4 can be any real number ( ) (, ) Write the expression under the radical. (The expression is called the radicand.) Since the radical is a cube root the expression can be any real number. Write the domain using interval notation. Evaluate expressions involving rational exponents For the problems in this group, an expression containing rational exponents should be written using radical notation, and an expression containing radical notation should be written using rational exponents. n n The definition of rational exponent is x x n The definition of a negative exponent is x. n x All rules for exponents apply to rational and negative exponents. Often used rules are listed below. Assume: a 0 m n m n Product Rule: a a a + m Power Rule: ( a ) n m a a mn m n Quotient Rule: a n a Example: Write 7 using rational exponents. The expression contains a cube root, which could be rewritten using rational exponents. 7 7 ( ) 7 7 * Rewrite the radical using a rational exponent. Use the power rule.
14 Page 4 of Example: Evaluate 8. The exponent is, which is a rational number. Use the definition for rational exponents. 8 8 (8 ) ( 8) () 4 Use the power rule. The definition of a rational exponent is used. Simplify the cube root. Note: If you use a calculator, remember to use parentheses. Enter 8^(/), not 8^/.
15 Page 5 of Example: Evaluate 9 The exponent is, which is a negative rational number ( 9 ) ( 9 ) ( ) 7 Use the definition of a negative exponent. Use the power rule. Write the power as square root. Simplify the square root and raise the result to the third power. If you use a calculator, remember to use parentheses. Enter 9^(/), not 9^/
16 Page 6 of Solve radical equations A radical equation is an equation containing one or more radical terms. For example, x x is a radical equation. To solve means to determine all the real values which, when substituted in the equation for x, will make the statement true. All such real values should be included in the answer. Note that 0 0 is true. Hence, x 0 is a solution of the equation x x. It turns out that x is also a solution of x x verify that This second solution is often ignored. To find all the solutions, follow the steps for solving radical equations given below. For equations containing one radical, the steps are:. Isolate the radical.. Square both sides of the equation. Solve the resulting equation which no longer contains radicals. This equation is often linear, quadratic, or rational. 4. Check the answers. (It is possible that some x values may be in the solution set of the resulting equation, but will not make the original equation true.) For equations with two radicals:. Isolate one of the radicals.. Square both sides of the equation.. Combine like terms. 4. Now the equation either has no radicals, or just one radical term. a. If there are no radicals follow steps and 4 under equations containing one radical. b. If there is one radical follow steps through 4 under equations containing one radical.
17 Page 7 of Example: Solve the equation x + 6 This is an equation (contains an equal sign). Use the steps for equations containing one radical. x + 6 ( x + ) 6 x + 6 x + 6 x 4 The radical is isolated. Square both sides. Solve the resulting linear equation. Check Substitute x 4 into the original equation and simplify the results. The statement is true so the solution is x 4. Example: Solve the equation x + 6 This is an equation (contains an equal sign). Use the steps for equations containing one radical. x + 6 ( x + ) ( 6) x + 6 x 4 The radical is isolated. Square both sides. Solve the resulting linear equation. Check Substitute x 4 into the original equation and simplify the results. The statement is false, so x 4 is not a solution. Note: The square root of any real number can t be negative. Hence, x + 6 can t be true.
18 Page 8 of Example: Solve the equation x + 4 x 40. This is an equation (contains an equal sign). It contains two radical terms. x + 4 x 40 5 x 40 x 8 ( ) x 8 x x.5 Add like radicals. Follow the steps for equations containing one radical. Divide both sides by 5. The radical is isolated. Square both sides. Solve the resulting linear equation. Check ( ) ( ) ( ) Substitute x into the original equation and simplify the results. 65 The statement is true so x is a solution to the equation.
19 Page 9 of Example: Solve the equation x x 0 This is an equation (contains an equal sign). It contains one radical term. Check x x 0 x x ( x ) ( x) x 4x x x 4 0 x( 4 x) 0 x 0 or ( 4x ) 0 x 0 or 4x x 0 or x 4 Check 0 Check x : ( ) x : Isolate the radical. Square both sides; remember to square each factor. This is a quadratic equation that can be solved by factoring. Set each factor equal to zero. Solve the two linear equations. There are two possible answers. Substitute each answer into the original equation. Simplify the results to be sure each yields a true statement. Both answers check.
20 Page 0 of Example: Solve the equation x + x This is an equation (contains an equal sign). It contains one radical term. x + x x + ( x ) x + x 6x + 9 x 7x x x 6 0 ( )( ) x 0 or x 6 0 x or x 6 Check Check x : + 4 Check x 6 : The radical is isolated. Square both sides. Note: ( x ) x. This is a quadratic equation that can be solved by factoring. Set each factor equal to zero. Solve the two linear equations. There are two possible solutions. Substitute each answer into the original equation. Simplify the results to be sure each yields a true statement. Notice x does not yield a true statement, while x 6 yields a true statement. The solution is x 6.
21 Page of Example: Solve the equation x + x 4 This is an equation (contains an equal sign). It contains two radical terms. Use the steps for equations that contain two radicals. x+ x 4 x ( x )( x ) ( x ) ( x ) ( ) 4 x x x x+ + x 4 x+ + 4 x x 4+ x 4 x 4 6 x 4 ( x ) 6 x 4 ( x ) x 4 x ( 4) x ( x 5) 0 or ( x 8) 0 x 5 or x 8 Isolate one radical by adding x 4 to both sides. Square both sides. + x 4 + x 4 Note: ( ) Now there is one radical term left in the equation. Combine like terms and isolate the radical. Factor two out of the terms on the right side. Divide both sides of the equation by a common factor of. Squared both sides. This is a quadratic equation that can be solved by factoring. Set each factor equal to zero. Solve the two linear equations. There are two possible solutions.
22 Page of Check Check 5 Check 8 x : ( ) x : ( ) Substitute each answer into the original equation. Simplify the results to be sure each yields a true statement. Both x 5 and x 8 check, so they are both solutions.
MAT 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 informationBasic Algebra Practice Test
1. Exponents and integers: Problem type 2 Evaluate. Basic Algebra Practice Test 2. Exponents and signed fractions Evaluate. Write your answers as fractions. 3. Exponents and order of operations Evaluate.
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 informationSolving ax 2 + bx + c = 0 Deriving the Quadratic Formula
Solving ax + bx + c = 0 SUGGESTED LEARNING STRATEGIES: Marking the Text, Group Presentation, Activating Prior Knowledge, Quickwrite Recall solving quadratic equations of the form a x + c = 0. To solve
More informationUnit 7: Quadratics Principles of Math 9
Unit 7: Quadratics Principles of Math 9 Lesson Topic Assignment 1 Solving by Square Roots Day #1 2 Solving by Factoring (Part 1) Day #2 3 Solving by Factoring (Part 2) Day #3 4 All Types of Factoring Day
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 informationALGEBRA I A PLUS COURSE OUTLINE
ALGEBRA I A PLUS COURSE OUTLINE 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 of Best
More informationMultiplying With Polynomials What do you do? 1. Distribute (or doubledistribute/foil, when necessary) 2. Combine like terms
Regents Review Session #1 Polynomials Adding and Subtracting Polynomials What do you do? 1. Add/subtract like terms Example: 1. (8x 39x 2 + 6x + 2)  (7x 35x 2 + 1x  8) Multiplying With Polynomials
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 information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationSimplifying 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 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 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 informationName: Date: Algebra 2/ Trig Apps: Simplifying Square Root Radicals. Arithmetic perfect squares: 1, 4, 9,,,,,,...
RADICALS PACKET Algebra 2/ Trig Apps: Simplifying Square Root Radicals Perfect Squares Perfect squares are the result of any integer times itself. Arithmetic perfect squares: 1, 4, 9,,,,,,... Algebraic
More information2. Simplify. College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses
College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2
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 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 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 informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
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 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 informationDevelopmental Math Course Outcomes and Objectives
Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/PreAlgebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and
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 informationAlgebra I Schedule MISSION FOUNDATIONS INTRODUCTION TO ALGEBRA. Order of operations with negative numbers. Combining like terms
Algebra I Schedule Week 2 MISSION FOUNDATIONS Order of operations with negative numbers Combining like terms Exponents with integer bases Square roots of perfect squares Simplify fractions Ordering rational
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 information= ( 2) 2 = x 2. x
Rationalizing the Denominator or Numerator Sometimes the denominator or numerator of a fraction has two terms and involves square roots, such as 3 5 or 2 + 3. The denominator or numerator may be rationalized
More informationBanking, Binomials defined, 344 multiplying (see Distributing) Box and whisker plot, Brackets and braces, 166
Absolute Value defined, 682 equations, 689690, 695697 and the number line, 684, 689 Addition 5860 of like terms, 316 of rational expressions, 249250 undoing, 1415 of x s, 125126 Algebra defined,
More informationElementary Algebra, 3rd Edition
Elementary Algebra, 3rd Edition by Robert H. Prior Tables of Contents (Updated 82011) Chapter 0, a PreAlgebra Review, is currently online only: http://bobprior.com/mat52ch0/mat52ch0.html Chapter 1
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 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 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 informationPRECALCULUS Semester I Exam Review Sheet
PRECALCULUS Semester I Exam Review Sheet Chapter Topic P.1 Real Numbers {1, 2, 3, 4, } Natural (aka Counting) Numbers {0, 1, 2, 3, 4, } Whole Numbers {, 3, 2, 2, 0, 1, 2, 3, } Integers Can be expressed
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationAlgebra 2 Correlation of the ALEKS course Algebra 2 to the Alabama Course of Study for Algebra II
Algebra 2 Correlation of the ALEKS course Algebra 2 to the Alabama Course of Study for Algebra II Number and Operations 1: Determine the relationships among the subsets of complex numbers. Integers and
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More informationAlgebra 1 Review for Algebra 2
for Algebra Table of Contents Section Topic Page 1.... 5. 6. Solving Equations Straightlined Graphs Factoring Quadratic Trinomials Factoring Polynomials Binomials Trinomials Polynomials Eponential Notation
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 informationRadical Expressions Squaring a # and finding the square root and are inverse operations. Cubing a # and finding the cube root are inverse operations.
Radical Expressions Squaring a # and finding the square root and are inverse operations. Since 5 2 = 25 a square root of 25 is 5. Since 5) 2 = 25, 5 is also a square root. Meaning if x 2 = 25 then x
More informationPrep for Intermediate Algebra
Prep for Intermediate Algebra This course covers the topics outlined below, new topics have been highlighted. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum
More informationEquations that Are Quadratic in Form
9.4 Equations that Are Quadratic in Form 9.4 OBJECTIVES 1. Solve a radical equation that is quadratic in form. Solve a fourth degree equation that is quadratic in form Consider the following equations:
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
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 informationIdentify examples of field properties: commutative, associative, identity, inverse, and distributive.
Topic: Expressions and Operations ALGEBRA II  STANDARD AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers
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 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 informationModuMath Algebra Lessons
ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations
More informationSect Solving and graphing inequalities
81 Sect 2.7  Solving and graphing inequalities Concepts #1 & 2 Graphing Linear Inequalities Definition of a Linear Inequality in One Variable Let a and b be real numbers such that a 0. A Linear Inequality
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 informationChapter Audio Summary for McDougal Littell PreAlgebra
Chapter Audio Summary for McDougal Littell PreAlgebra Chapter Rational Numbers and Equations In Chapter you learned to write, compare, and order rational numbers. Then you learned to add and subtract
More informationKEY. i is the number you can square to get an answer of 1. i 2 = 1. Or, i is the square root of 1 i =
Real part Imaginary part Algebra Unit: 05 Lesson: 0 Solving Quadratic Equations by Formula a _ bi KEY Complex Numbers All the quadratic equations solved to this point have had two real solutions or roots.
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 informationUnit 7: Radical Functions & Rational Exponents
Date Period Unit 7: Radical Functions & Rational Exponents DAY 0 TOPIC Roots and Radical Expressions Multiplying and Dividing Radical Expressions Binomial Radical Expressions Rational Exponents 4 Solving
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationMonomials. Polynomials. Objectives: Students will multiply and divide monomials Students will solve expressions in scientific notation
Students will multiply and divide monomials Students will solve expressions in scientific notation 5.1 Monomials Many times when we analyze data we work with numbers that are very large. To simplify these
More informationChapter 1: Number Systems and Fundamental Concepts of Algebra. If n is negative, the number is small; if n is positive, the number is large
Final Exam Review Chapter 1: Number Systems and Fundamental Concepts of Algebra Scientific Notation: Numbers written as a x 10 n where 1 < a < 10 and n is an integer If n is negative, the number is small;
More informationChapter 9: Quadratic Functions 9.3 SIMPLIFYING RADICAL EXPRESSIONS
Chapter 9: Quadratic Functions 9.3 SIMPLIFYING RADICAL EXPRESSIONS Vertex formula f(x)=ax 2 +Bx+C standard d form X coordinate of vertex is Use this value in equation to find y coordinate of vertex form
More informationUnit 1, Review Transitioning from Previous Mathematics Instructional Resources: Prentice Hall: Algebra 1
Unit 1, Review Transitioning from Previous Mathematics Transitioning from Seventh grade mathematics to Algebra 1 Read, compare and order real numbers Add, subtract, multiply and divide rational numbers
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 informationPrep for College Algebra
Prep for College Algebra This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet
More informationAlgebra 1 Key Vocabulary Words Chapter 11
Section : radical expression radical function square root function parent square root function Section 11.2: radical expression (see Ch ) simplest form of a radical expression rationalizing the denominator
More informationMath 1111 Journal Entries Unit I (Sections , )
Math 1111 Journal Entries Unit I (Sections 1.11.2, 1.41.6) Name Respond to each item, giving sufficient detail. You may handwrite your responses with neat penmanship. Your portfolio should be a collection
More informationChapter R  Basic Algebra Operations (69 topics, due on 05/01/12)
Course Name: College Algebra 001 Course Code: R3RK6CTKHJ ALEKS Course: College Algebra with Trigonometry Instructor: Prof. Bozyk Course Dates: Begin: 01/17/2012 End: 05/04/2012 Course Content: 288 topics
More informationAlgebra II Pacing Guide First Nine Weeks
First Nine Weeks SOL Topic Blocks.4 Place the following sets of numbers in a hierarchy of subsets: complex, pure imaginary, real, rational, irrational, integers, whole and natural. 7. Recognize that the
More informationAlgebra Revision Sheet Questions 2 and 3 of Paper 1
Algebra Revision Sheet Questions and of Paper Simple Equations Step Get rid of brackets or fractions Step Take the x s to one side of the equals sign and the numbers to the other (remember to change the
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 informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationSimplifying Expressions with Zero and Negative Exponents
Simplifying Expressions with Zero and Negative Exponents How are exponential functions used to model change? Lesson Title Objectives Standards Simplifying Expressions with Zero and Negative Exponents (7.1)
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 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 informationSummer Review Packet For Algebra 2 CP/Honors
Summer Review Packet For Algebra CP/Honors Name Current Course Math Teacher Introduction Algebra builds on topics studied from both Algebra 1 and Geometry. Certain topics are sufficiently involved that
More informationUNIT ONE RADICALS 15 HOURS MATH 521B
UNIT ONE RADICALS 15 HOURS MATH 521B Revised Nov 9, 00 19 SCO: By the end of grade 11 students will be expected to: A demonstrate an understanding of the role of irrational numbers in applications Elaboration
More informationMAC COURSE OBJECTIVES
MAC 1105  COURSE OBJECTIVES Your Homework problems are the online problems available at the russell egrade site. (http://russell.math.fsu.edu:9888/) There is no required text for MAC 1105. The sections
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 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 information1.6 The Order of Operations
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
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 informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationUnit #8 Radicals and the Quadratic Formula
Unit #8 Radicals and the Quadratic Formula Lessons: 1  Square Root Functions 2  Solving Square Root Equations  The Basic Exponent Properties 4  Fractional Exponents Revisited 5  More Exponent Practice
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 informationALGEBRA 2 SCOPE AND SEQUENCE. Writing Quadratic Equations Quadratic Regression
ALGEBRA 2 SCOPE AND SEQUENCE UNIT 1 2 3 4 5 6 DATES & NO. OF DAYS 8/229/16 19 days 9/199/30 9 days 10/310/14 9 days 10/2010/25 4 days 10/2712/2 17 days 12/512/16 10 days UNIT NAME Foundations for
More informationList of MAT 099 Mathematical Concept Practice Sheets. (updated Spring 15)
List of MAT 099 Mathematical Concept Practice Sheets (updated Spring 15) Chapter 1: Linear Equations and Inequalities 1. Multiplying and Dividing Sign Numbers 2. Basics of Exponents 3. Adding and Subtracting
More informationPrentice Hall Mathematics, Algebra 2, Indiana Edition 2011
Prentice Hall Mathematics, Algebra 2, Algebra 2 C O R R E L A T E D T O from March 2009 Algebra 2 A2.1 Functions A2.1.1 Find the zeros, domain, and range of a function. A2.1.2 Use and interpret function
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 informationAccuplacer Elementary Algebra Study Guide for Screen Readers
Accuplacer Elementary Algebra Study Guide for Screen Readers The following sample questions are similar to the format and content of questions on the Accuplacer Elementary Algebra test. Reviewing these
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
More informationGlencoe/McGrawHill. Algebra Alabama Course of Study: Algebra II ISBN# correlated to
Glencoe/McGrawHill Algebra 2 2003 ISBN# 0 02 827999 2 correlated to Alabama Course of Study: Algebra II Number and Operations Students will: 1. Determine the relationships among the subsets of complex
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 informationCOMPLEX NUMBERS. Algebra 2 & Trigonometry
COMPLEX NUMBERS Algebra & Trigonometry Name: Topic Pages Day 1 Imaginary Numbers / Powers of i 3 4 Day Graphing and Operations with Complex Numbers 8 13 Day 3 Dividing Complex Numbers 14 17 Day 4 Complex
More informationELEMENTARY ALGEBRA. Practice Questions. Order of Operations
ELEMENTARY ALGEBRA Overview The Elementary Algebra section of ACCUPLACER contains 12 multiple choice Algebra questions that are similar to material seen in a PreAlgebra or Algebra I precollege course.
More informationCurriculum Map
ASSESSMENTS SEPTEMBER What is a numeric expression? What is a variable? How do you simplify variable expressions? How do you evaluate algebraic expressions? What is the distributive property? How do you
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
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 informationMath 2200 Chapter 1 Arithmetic and Geometric Sequences and Series Review
Math 00 Chapter 1 Arithmetic and Geometric Sequences and Series Review Key Ideas Description or Example Sequences Series Arithmetic Sequence An ordered list of numbers where a mathematical pattern can
More informationLesson 71. Roots and Radicals Expressions
Lesson 71 Roots and Radicals Epressions Radical Sign inde Radical Sign n a Radicand Eample 1 Page 66 #6 Find all the real cube roots of 0.15 0.15 0.15 0.15 0.50 (0.50) 0.15 0.50 is the cube root of 0.15.
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationReview for College Algebra Midterm 1
Review for College Algebra Midterm 1 1.2 Functions and graphs Vocabulary: function, graph, domain, range, VerticalLine Test. Determine whether a given correspondence/mapping is a function. Evaluate a
More informationMath 002 Intermediate Algebra
Math 002 Intermediate Algebra Student Notes & Assignments Unit 4 Rational Exponents, Radicals, Complex Numbers and Equation Solving Unit 5 Homework Topic Due Date 7.1 BOOK pg. 491: 62, 64, 66, 72, 78,
More informationTable of Contents Sequence List
Table of Contents Sequence List 368102215 Level 1 Level 5 1 A1 Numbers 010 63 H1 Algebraic Expressions 2 A2 Comparing Numbers 010 64 H2 Operations and Properties 3 A3 Addition 010 65 H3 Evaluating
More information