EXAMPLE EXAMPLE. Simplify. Simplify each expression. See left. EXAMPLE Real-World Problem Solving EXAMPLE. Write = xa1 1!5 B = 162 Cross multiply.
|
|
- Tracey Craig
- 6 years ago
- Views:
Transcription
1 -. Plan Lesson Preview Check Skills You ll Need Operations With Radical Epressions Lesson -: Eamples,, 7 Eercises, Etra Practice, p. 7 Lesson Preview What You ll Learn - To simplify sums and differences To simplify products and quotients... And Why To find the width of a painting, as in Eample Operations With Radical Epressions Check Skills You ll Need (For help, go to Lesson -.) Simplify each radical epression..! ".!00 0".! ". " Rationalize each denominator. "! "! "0! "0.. 7.!!8! New Vocabulary like radicals unlike radicals conjugates Lesson Resources Teaching Resources Practice, Reteaching, Enrichment Reaching All Students Practice Workbook - Spanish Practice Workbook - Basic Algebra Planning Guide - Presentation Assistant Plus! Transparencies Check Skills You ll Need - Additional Eamples - Student Edition Answers - Lesson Quiz - PH Presentation Pro CD - Computer Test Generator CD Technology Resource Pro CD-ROM Computer Test Generator CD Prentice Hall Presentation Pro CD Student Site Teacher Web Code: aek-00 Self-grading Lesson Quiz Teacher Center Lesson Planner Resources Plus Part Simplifying Sums and Differences Check Understanding Need Help? Multiplication Property of Square Roots:!ab =!a?!b and!a?!b =!ab Check Understanding For radical epressions, like radicals have the same radicand. Unlike radicals do not have the same radicand. For eample,!7 and!7 are like radicals, but! and! are unlike radicals. To simplify sums and differences, you use the Distributive Property to combine like radicals. Simplify! +!. Combining Like Radicals! +! =! +! Both terms contain!. = ( + )! Use the Distributive Property to combine like radicals. =! Simplify. Simplify each epression. a.! -! 7 " b.!0 -!0 "0 You may need to simplify a radical epression to determine if you have like radicals. Simplify 7! -!. Simplifying to Combine Like Radicals Interactive lesson includes instant self-check, tutorials, and activities. 7! -! = 7! -!? is a perfect square and a factor of. = 7! -!?! Use the Multiplication Property of Square Roots. = 7! -! Simplify!. = (7 )! Use the Distributive Property to combine like radicals. =! Simplify. Simplify each epression. a.!0 +! 8 " b.! -!7 " Chapter Radical Epressions and Equations Ongoing Assessment and Intervention Before the Lesson Diagnose prerequisite skills using: Check Skills You ll Need During the Lesson Monitor progress using: Check Understanding Additional Eamples Standardized Test Prep After the Lesson Assess knowledge using: Lesson Quiz Computer Test Generator CD
2 Part Simplifying Products and Quotients. Teach When simplifying a radical epression like! A! 7B, use the Distributive Property to multiply! times A! 7B. Simplify!A! 7B. Using the Distributive Property!A! 7B =!8 + 7! Use the Distributive Property. =!9?! + 7! Use the Multiplication Property of Square Roots. =! + 7! Simplify. Math Background Unlike radicals, such as! and!, are analogous to different variables, such as and y. They cannot be combined by adding or subtracting. Teaching Notes Check Understanding Check Understanding Need Help? Remember that the difference of two squares can be factored as (a + b)(a - b). Simplify each radical epression. a.!a!0b b.!a! B c. If both radical epressions have two terms, you can multiply the same way you find the product of two binomials, by using FOIL. Simplifying Using FOIL Simplify A! -!BA! +!B. A! -!BA! +!B =! +!7 -!7 -! Use FOIL. = -!7 - () Combine like radicals and simplify! and!. = -!? - 0 is a perfect square factor of 7. = -!?! - 0 Use the Multiplication Property of Square Roots. = -! - 0 Simplify!. =- -! Simplify. Simplify each radical epression. a. A! +!BA! -!B b. A!7 + B!aA!a B " ± " " " a ± "a " ± 8"7 Conjugates are the sum and the difference of the same two terms. The radical epressions! +! and! -! are conjugates. The product of two conjugates results in a difference of two squares. A!!BA!!B = A!B - A!B = - = Notice that the product of these conjugates has no radical. You recall that a simplified radical epression has no radical in the denominator. When a denominator contains a sum or a difference including radical epressions, you can rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. For eample, to simplify a radical epression like, you multiply by " ". " " " " Lesson - Operations With Radical Epressions 0 Technology Tip Suggest that students use a calculator to verify that the combining of like terms is correct. Find the value of the original equation and compare it to the value of the answer. Additional Eamples Simplify! +!.! Simplify 8! -!.! Teaching Notes Inclusion Students with some kinds of vision problems will have difficulty distinguishing between the terms, especially when using FOIL, due to the distraction of so many radical signs. Suggest that they add etra space between the epressions and around each operation sign, and use arcs to connect each pair of terms to be multiplied. Additional Eamples Simplify! (!8 + 9).!0 ± 9! Simplify (! -!)(! +!). 7! Reaching All Students Below Level To emphasize to students that only like radicals can be combined show them!9 +!!. Rather!9 +! = + = 7. Advanced Learners Have students simplify. " "y English Learners See note on page 0. Inclusion See note on page 0. 0
3 Additional Eamples 8 Simplify.!7 ±! Á 7 Á The ratio length : width of a painting is approimately equal to the golden ratio ( +!). The length of the painting is in. Find the eact width of the painting in simplest radical form. Then find the approimate width to the nearest inch. (Á ) in.; in. English Learners The letters j and g are pronounced differently in different languages. Help students pronounce the word conjugate. Eplain that conjugate means to join together in pairs. Connection to Art Encourage students to research how the golden ratio was used in Renaissance architecture. Invite interested students to make a presentation for the class describing the golden ratio and showing pictures of buildings incorporating the golden ratio proportions. Closure Ask students to summarize how to add and subtract radicals. You remove perfect squares that are factors of the radicands, and write their square roots outside the radical signs and combine like terms. "7 "0 " 8 " 8 " " Check Understanding Simplify.!!!! Rationalizing a Denominator Using Conjugates =!!? Multiply the numerator and the denominator by the conjugate of the denominator. A!!B = Multiply in the denominator. A!! B = Simplify the denominator. = A!!B Divide and by the common factor. =!! Simplify the epression. Simplify each epression. See left. a. b. c.!7!!0!8 You can solve a ratio involving radical epressions. Real-World Problem Solving Art The ratio length : width of this painting by Mondrian is approimately equal to the golden ratio A +!B i. The length of the painting is 8 inches. Find the width of the painting in simplest radical form. Then find the approimate width to the nearest inch. Define 8 = length of painting = width of painting Relate A! B i = length i width! Write = 8 A! B = Cross multiply. A! B = Divide both sides by A! B. A! B A! B Multiply the numerator and the = A! B? A!B A!B denominator by the conjugate of the denominator. A!B = Multiply in the denominator. A!B = Simplify the denominator. 8A!B = Divide and by the common factor. Use a calculator. = < 0!!!!!! The eact width of the painting is painting is 0 inches. 8A!B inches. The approimate width of the Check Understanding Another painting has a length i width ratio approimately equal to the golden ratio A +!B i. Find the length of a painting if the width is inches. in. 0 Chapter Radical Epressions and Equations pages 0 0 Eercises. 9 ". 8 0"0. "7. ± "0. ± 9" 7. " 0
4 EXERCISES Practice and Problem Solving A B Practice by Eample Eample (page 00) Eample (page 00) Eample (page 0) Eample (page 0) Eample (page 0) Eample (page 0) Apply Your Skills Simplify each epression..! + 8! ".!0 +!0 8"0.! -! " Tell whether each pair of epressions can be simplified to like radicals. 7.!,! yes 8.!,!7 yes 9.!,!0 no Simplify each epression. 0.!8 +! ".! - 7! ".!8 +! ".! -! ".!7 -!8 "7.!0 +!0 8"0.!A!8 - B " 7.!A!7 + B 9 ± " 8.!A! - B " 9.!A! + B " ± " 0.!A +!B " ±.!A! - B ". A! +!BA! -!B. A! -!BA! -!B See margin p. 0. A!7 B. A!0 +!B. A! + BA! + B 7. A -!BA9 +!B 8. 8 "7 ± " 9.!7!!8! " "0 " 0. 8 " ".!!8!0!. 0 " ". 9 8 " ± 9"!!!! Find an eact solution for each equation. Find the approimate solution to the nearest tenth.. See margin.!. =.!! =. =!!!! 7. The ratio of the length to the width of a painting is A! B i. The length is ft. What is the width? 7. ft Simplify each epression. 9. See margin. For more practice, see Etra Practice..!7 -!7 "7.! -! ".! -! 8" 8.!0 +!90 "0 9.!A +!B 0.! +!7 -!. A!!0!! B.. A!7 +!8BA!7 +!8B!!!!.!A! +!8B.!0-7!8.!9!. Practice Assignment Guide Objective A Objective A C B B Core, 8, Core 7, Etension 7 Standardized Test Prep 7 7 Mied Review 7 9 Error Prevention Eercises 0 Students may think a radical cannot be simplified because they choose factors that do not contain a perfect square. For eample, for!8 students might choose and instead of 9 and. Suggest to them that they factor the radicand completely and look for pairs of factors. Alternative Method Eercises 0 7 Help students see that when multiplying the square roots of identical radicands, the product is just the radicand. This method is quicker than multiplying the radicands, and then finding the square root of the product. Enrichment - Reteaching - Practice - Name Class Date Practice - Simplify each epression.. "7 + "7. 0" - ". Operations with Radical Epressions "Q "R " " " ". "Q" "R 7. Chemistry The ratio of the rates of diffusion of two gases is given by the r formula!m = r, where m and m are the masses of the molecules of the!m r gases. Find if m = units and m = 0 units. "0 r Lesson - Operations With Radical Epressions 0. 0( " ± );. 0. ".. ;.. 8 ± ". " " "0 ". "; " ± " ± " ± 9. " ± ". ± " 7. "8 + " 8. " - 8" 9. "Q" "R 0. "8 - "0. " + "8. " ". Q8" 7R. 8Q" "R. 7" - ". "Q7 "R 7. 8Q "R 8. " + " " + " 0. 8 " + 0". "0Q "R. 9" - "0. 0" - 7". " - ". "7 + "8. 8" - " 7. "0 + "0 8. "Q" "R 9. "9 - "9 0. 0" - ". 8" - "7. "Q" "R. 7 " + ". "9 + "9. "9 - "9. "8Q" 7R " " "7 " " 0. Q".. 7 R Q" "R " "7 ".. " ". " " " " Solve each eercise by using the golden ratio Q "R :.. The ratio of the height ; width of a window is equal to the golden ratio. The width of the door is in. Find the height of the door. Epress your answer in simplest radical form and in inches. 7. The ratio of the length ; width of a flower garden is equal to the golden ratio. The width of the garden is ft. Find the length of the garden. Epress your answer is simplest radical form and in feet. 8. The ratio of the width ; height of the front side of a building is equal to the golden ratio. The height of the building is 0 ft. Find the width of the building. Epress your answer in simplest radical form and in feet. Lesson - Practice Algebra Chapter Pearson Education, Inc. All rights reserved. 0
5 Connection to History Eercise The greatest number of kites flown on a single line is,8. Sadao Harada and a team of assistants achieved this feat in Kagoshima, Japan, in October 990. Math Tip Eercises Remind students that for some kinds of problems percents must be rounded up, disregarding the usual rounding rules. Show them that if they round down they will not reach the target amount. 8. 8" units 9. (0 ± 0 ") units 0. "0 units. ( ± "0) units Geometry Find the eact perimeter of each figure below. 8. See left. 8. y 9. y 0.. O O 0 a. The student simplified "8 as " instead of " or ". b. " ± " a. " or.8 ft. Open-Ended Make up three sums that are less than or equal to 0. Use the square roots of,,, or 7, and the whole numbers less than 0. For eample, 8" 9"7 # 0. See margin.. Error Analysis When simplifying " "8, a student wrote " ". a. What error did the student make? See left. b. Simplify " "8 correctly.. You can make a bo kite like the one at the right in the shape of a rectangular solid. The opening at each end of the kite is a square. a. Suppose the sides of the square are ft long. How long are the diagonal struts used for bracing? See left. b. Suppose each side of the square has length s. Find the length of the diagonal struts in terms of s. Write your answer in simplest form. s" Investments For Eercises 7, the formula r Î A P gives the interest rate r that will allow principal P to grow into amount A in two years, if the interest is compounded annually. Use the formula to find the interest rate you would need to meet each goal.. Suppose you have $00 to deposit into an account. Your goal is to have $9 in that account at the end of the second year. 9.%. Suppose you have $0 to deposit into an account. Your goal is to have $700 in that account at the end of two years..8% 7. Suppose you have $00 to deposit into an account. Your goal is to have $800 in that account at the end of two years..% 8. a. Suppose n is an even number. Simplify " n. b. Suppose n is an odd number greater than. Simplify " n. a"b 9. Critical Thinking Simplify. "ab b"a b n s n " s 0 0 Chapter Radical Epressions and Equations pages 0 0 Eercises. Answers may vary. Sample: 8 " ± ", "7 ± 9 ", " ± "7
6 0. Find the value of the numerical epression for Professor Hinkle s age in the cartoon. about years. Assess b. No; the only values it worked for were 0 and. They are unlike radicands.. Writing Eplain why! +! cannot be simplified.. a. Copy and complete the table. a b a b a b a b 0 0 "7 9 8 " "8 b. Does!a +!b always equal!a b? Eplain. See left.. Error Analysis Eplain the error in the work below.! =! =! +! = + = 9 "a b u "a ± "b Lesson Quiz - Simplify each epression..! -! 0.!0 -!!.!(! +!) ±!. (! -!)(! +!) ±!7. 8! 8!7 Á Á 7 Alternative Assessment Divide the class into small groups. Tell students they will be team teachers who will teach a class what they should learn in this lesson. Have the group design a problem for each of Eamples and present to the class the problems and how to solve them. Allow groups to use the board. C Challenge Simplify each epression. 9"!8 "7.! !!7 Å 8"!7!8! !88 +!0 -!98 0"! Å 70. " " " ± 9. A! +!BA! +!8 +!B Find the length of each hypotenuse. Write your answers in simplified radical form. a. b. 0!!0 "!0! 0 0 " 0 " c. If the length of the legs of a right triangle are!p +!q and!p -!q, write an epression for the length of the hypotenuse. "(p q) Standardized Standardized Test Prep Test Prep Multiple Choice 7. Simplify!7 +!7. B A.! B.! C.!0 D.!0 Lesson - Operations With Radical Epressions 0 0
7 Standardized Test Prep Resources For additional practice with a variety of test item formats: Standardized Test Prep, p. Test-Taking Strategies, p. 8 Test-Taking Strategies with Transparencies Eercise 7 Remind students to use FOIL. Encourage them to draw curved arrows from each term in the first binomial to each term in the second binomial to help in multiplying the terms together. Take It to the NET Online lesson quiz at Web Code: aea-0 Short Response Etended Response Mied Review Lesson - Lesson 0- Lesson 9-7. Which radical epression is NOT equal to "? I F.!8 +!8 G.!98 -!8 H.! +! I.!8 +! 7. Simplify A! -! B A! +! B. Show your work. See back of book. 7. Eplain the steps needed to simplify. See back of book.!7! Find the distance between the points in each pair. If necessary, round to the nearest tenth. 7. (, ), (8, ) 9. units 77. (-, 7), (, 0).7 units 78. (-, ), (0, -). units Find the midpoint of each segment with the given endpoints. 79. A(, -) and B(, ) (, ) 80. H(-, ) and K(, 7) (,.) Solve each equation by factoring. 8. t - t = 0 0, 7 8. p, 9-7p - 8 = 0 8. k 9, + k + 7 = 0 8. y - y =, 8. m + 0 =-7m 8. a =-7a -, Find each product See margin., 87. (b + )(b + ) 88. (p + 7)(p + 7) 89. (g - 7)(g + 7) 90. ( + )( - ) 9. k 9R Q k 9R 9. (d -.)(d -.) 9 9k 8 d.d ±. Algebra at Work Auto Mechanic Auto mechanics work to see that car engines get the most out of every gallon of gasoline. Formulas used by mechanics often involve radicals. For eample, a car gets its power when gas and air in each cylinder are compressed and ignited by a spark plug. An engine s efficiency e is given c "c by the formula e = c, where c is the compression ratio. Because of the compleity of such formulas and of modern highperformance engines, today s auto mechanic must be a highly trained and educated professional who understands algebra, graph reading, and the operation of computerized equipment. Take It to the NET For more information about a career as an auto mechanic, go to Web Code: aeb-0 0 Chapter Radical Epressions and Equations pages 0 0 Eercises 87. b ± b ± 88. p ± 8p ± g 9 0
Simplifying a Rational Expression. Factor the numerator and the denominator. = 1(x 2 6)(x 2 1) Divide out the common factor x 6. Simplify.
- Plan Lesson Preview Check Skills You ll Need Factoring ± ± c Lesson -5: Eamples Eercises Etra Practice, p 70 Lesson Preview What You ll Learn BJECTIVE - To simplify rational epressions And Why To find
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 information150. a. Clear fractions in the following equation and write in. b. For the equation you wrote in part (a), compute. The Quadratic Formula
75 CHAPTER Quadratic Equations and Functions Preview Eercises Eercises 8 50 will help you prepare for the material covered in the net section. 8. a. Solve by factoring: 8 + - 0. b. The quadratic equation
More informationWhat You ll Learn. or irrational. New Vocabulary perfect square, square root, irrational numbers, real numbers. Why Learn This?
-. Plan - Exploring Square Roots and Irrational Numbers Objective To find and estimate square roots and to classify numbers as rational or irrational Examples Finding Square Roots of Perfect Squares Estimating
More informationNew Vocabulary equivalent inequalities. x 1 4, 7 and x, 3 are equivalent inequalities.
-. Plan - Solving Inequalities Using Addition and Subtraction Objectives To use addition to solve To use subtraction to solve Eamples Using the Addition Property of Inequality Solving and Checking Solutions
More informationSection 5.1 Extra Practice
Name: Date: Section.1 Etra Practice BLM 1. Epress each radical as a simplified mied radical. 0 98, 0 6 y, 0, y 0. Epress each mied radical as an equivalent entire radical. 1 9, 0 y 7 y, 0, y 0. Order each
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 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 informationProperties of Radicals
9. Properties of Radicals Essential Question How can you multiply and divide square roots? Operations with Square Roots Work with a partner. For each operation with square roots, compare the results obtained
More information8.2 Solving Quadratic Equations by the Quadratic Formula
Section 8. Solving Quadratic Equations by the Quadratic Formula 85 8. Solving Quadratic Equations by the Quadratic Formula S Solve Quadratic Equations by Using the Quadratic Formula. Determine the Number
More informationCircles in the Coordinate Plane. Find the length of each segment to the nearest tenth y. Distance Formula Square both sides.
-5 ircles in the oordinate Plane -5. Plan What You ll Learn To write an equation of a circle To find the center and radius of a circle... nd Wh To describe the position and range of three cellular telephone
More informationMini-Lecture 7.1 Radicals and Radical Functions
Mini-Lecture 7. Radicals and Radical Functions Learning Objectives:. Find square roots.. Approimate roots.. Find cube roots.. Find n th roots.. Find n a n when a is an real number. 6. Graph square and
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 #33 Solving Incomplete Quadratics
Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique
More information5-7 The Pythagorean Theorem
5-7 The Pythagorean Theorem Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Classify each triangle by its angle measures. 1. 2. acute right 3. Simplify 12 4. If a = 6, b = 7, and c = 12, find
More information3.1 Solving Quadratic Equations by Taking Square Roots
COMMON CORE -8-16 1 1 10 8 6 0 y Locker LESSON.1 Solving Quadratic Equations by Taking Square Roots Name Class Date.1 Solving Quadratic Equations by Taking Square Roots Essential Question: What is an imaginary
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 informationAdding and Subtracting Rational Expressions
COMMON CORE Locker LESSON 9.1 Adding and Subtracting Rational Epressions Name Class Date 9.1 Adding and Subtracting Rational Epressions Essential Question: How can you add and subtract rational epressions?
More informationSkills Practice Skills Practice for Lesson 3.1
Skills Practice Skills Practice for Lesson.1 Name Date Get Radical or (Be)! Radicals and the Pythagorean Theorem Vocabulary Write the term that best completes each statement. 1. An expression that includes
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 informationHow can you find decimal approximations of square roots that are not rational? ACTIVITY: Approximating Square Roots
. Approximating Square Roots How can you find decimal approximations of square roots that are not rational? ACTIVITY: Approximating Square Roots Work with a partner. Archimedes was a Greek mathematician,
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 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 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 informationModule 2, Section 2 Solving Equations
Principles of Mathematics Section, Introduction 03 Introduction Module, Section Solving Equations In this section, you will learn to solve quadratic equations graphically, by factoring, and by applying
More informationPatterns and Functions. Write an algebraic expression for each phrase more than twice a number 2. a number divided by 4
- Patterns and Functions -. Plan What You ll Learn To write a function rule To understand relationships of quantities in a function... And Why To find reasonable domain and range for real-world situations,
More informationNOTES: Chapter 11. Radicals & Radical Equations. Algebra 1B COLYER Fall Student Name:
NOTES: Chapter 11 Radicals & Radical Equations Algebra 1B COLYER Fall 2016 Student Name: Page 2 Section 3.8 ~ Finding and Estimating Square Roots Radical: A symbol use to represent a. Radicand: The number
More informationThe Pythagorean Theorem and Its Converse
The and Its onverse Use the. Use the converse of the. Vocabulary Pythagorean triple Study Tip Look ack To review finding the hypotenuse of a right triangle, see Lesson 1-3. are right triangles used to
More informationName Class Date. You can use the properties of equality to solve equations. Subtraction is the inverse of addition.
2-1 Reteaching Solving One-Step Equations You can use the properties of equality to solve equations. Subtraction is the inverse of addition. What is the solution of + 5 =? In the equation, + 5 =, 5 is
More informationAlgebra 2-2nd Semester Exam Review 11
Algebra 2-2nd Semester Eam Review 11 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine which binomial is a factor of. a. 14 b. + 4 c. 4 d. + 8
More informationEquations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE.1 The Rectangular Coordinate Systems and Graphs. Linear Equations in One Variable.3 Models and Applications. Comple Numbers.5 Quadratic Equations.6 Other
More informationMultiplying and Dividing Rational Expressions
6.3 Multiplying and Dividing Rational Epressions Essential Question How can you determine the ecluded values in a product or quotient of two rational epressions? You can multiply and divide rational epressions
More informationWrite in simplest form. New Vocabulary rate of change
-. Plan bjectives To find rates of change from tables and graphs To find slope Eamples Finding Rate of Change Using a Table Finding Rate of Change Using a Graph Finding Slope Using a Graph Finding Slope
More informationSolving Multi-Step Inequalities
- What You ll Learn To solve multi-step inequalities with variables on one side To solve multi-step inequalities with variables on both sides... And Why To find the measurements of a banner, as in Eample
More informationMathematics. Standards Plus. Grade COMMON CORE INTERVENTION SAMPLER
Mathematics Standards Plus COMMON CORE INTERVENTION Grade 7 SAMPLER Standards Plus COMMON CORE INTERVENTION Available for Grades 1-8 Language Arts and Math Standards Plus COMMON CORE INTERVENTION Mathematics
More informationMini-Lecture 5.1 Exponents and Scientific Notation
Mini-Lecture.1 Eponents and Scientific Notation Learning Objectives: 1. Use the product rule for eponents.. Evaluate epressions raised to the zero power.. Use the quotient rule for eponents.. Evaluate
More informationActivity 1 Multiply Binomials. Activity 2 Multiply Binomials. You can use algebra tiles to find the product of two binomials.
Algebra Lab Multiplying Polynomials You can use algebra tiles to find the product of two binomials. Virginia SOL A..b The student will perform operations on polynomials, including adding, subtracting,
More informationSpecial Right Triangles
. Special Right Triangles Essential Question What is the relationship among the side lengths of - - 0 triangles? - - 0 triangles? Side Ratios of an Isosceles Right Triangle ATTENDING TO PRECISION To be
More informationShenandoah University. (PowerPoint) LESSON PLAN *
Shenandoah University (PowerPoint) LESSON PLAN * NAME DATE 10/28/04 TIME REQUIRED 90 minutes SUBJECT Algebra I GRADE 6-9 OBJECTIVES AND PURPOSE (for each objective, show connection to SOL for your subject
More informationMATH 108 REVIEW TOPIC 6 Radicals
Math 08 T6-Radicals Page MATH 08 REVIEW TOPIC 6 Radicals I. Computations with Radicals II. III. IV. Radicals Containing Variables Rationalizing Radicals and Rational Eponents V. Logarithms Answers to Eercises
More informationFinding Complex Solutions of Quadratic Equations
COMMON CORE y - 0 y - - 0 - Locker LESSON 3.3 Finding Comple Solutions of Quadratic Equations Name Class Date 3.3 Finding Comple Solutions of Quadratic Equations Essential Question: How can you find the
More informationGeometry Honors Summer Packet
Geometry Honors Summer Packet Hello Student, First off, welcome to Geometry Honors! In the fall, we will embark on an eciting mission together to eplore the area (no pun intended) of geometry. This packet
More information6-5 Study Guide and Intervention
6-5 Study Guide and Intervention Simplify Radicals Product Property of Radicals For any real numbers a and b, and any integer n > 1: 1. if n is even and a and b are both nonnegative, then n ab n a n b.
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 informationEssential Question: How can you solve equations involving variable exponents? Explore 1 Solving Exponential Equations Graphically
6 7 6 y 7 8 0 y 7 8 0 Locker LESSON 1 1 Using Graphs and Properties to Solve Equations with Eponents Common Core Math Standards The student is epected to: A-CED1 Create equations and inequalities in one
More informationSYSTEMS OF THREE EQUATIONS
SYSTEMS OF THREE EQUATIONS 11.2.1 11.2.4 This section begins with students using technology to eplore graphing in three dimensions. By using strategies that they used for graphing in two dimensions, students
More informationObjectives To solve quadratic equations using the quadratic formula To find the number of solutions of a quadratic equation
9-6 The Quadratic Formula and the Discriminant Content Standards A.REI..a Use the method of completing the square to transform an quadratic equation in into an equation of the form ( p) 5 q... Derive the
More informationLooking Ahead to Chapter 10
Looking Ahead to Chapter Focus In Chapter, you will learn about polynomials, including how to add, subtract, multiply, and divide polynomials. You will also learn about polynomial and rational functions.
More informationMultiplying Polynomials. The rectangle shown at the right has a width of (x + 2) and a height of (2x + 1).
Page 1 of 6 10.2 Multiplying Polynomials What you should learn GOAL 1 Multiply two polynomials. GOAL 2 Use polynomial multiplication in real-life situations, such as calculating the area of a window in
More informationWrite your answers on notebook paper. Show your work.
UNIT 6 Getting Ready Use some or all of these exercises for formative evaluation of students readiness for Unit 6 topics. Prerequisite Skills Finding the length of the sides of special right triangles
More informationMini-Lecture 8.1 Solving Quadratic Equations by Completing the Square
Mini-Lecture 8.1 Solving Quadratic Equations b Completing the Square Learning Objectives: 1. Use the square root propert to solve quadratic equations.. Solve quadratic equations b completing the square.
More information8 th Grade Intensive Math
8 th Grade Intensive Math Ready Florida MAFS Student Edition August-September 2014 Lesson 1 Part 1: Introduction Properties of Integer Exponents Develop Skills and Strategies MAFS 8.EE.1.1 In the past,
More information6.2 Multiplying Polynomials
Locker LESSON 6. Multiplying Polynomials PAGE 7 BEGINS HERE Name Class Date 6. Multiplying Polynomials Essential Question: How do you multiply polynomials, and what type of epression is the result? Common
More informationWhy? 2 3 times a week. daily equals + 8_. Thus, _ 38 or 38% eat takeout more than once a week. c + _ b c = _ a + b. Factor the numerator. 1B.
Then You added and subtracted polynomials. (Lesson 7-5) Now Add and subtract rational epressions with like denominators. 2Add and subtract rational epressions with unlike denominators. Adding and Subtracting
More informationWhich Mathematics Course Should You Take? August 22, 2018 Which mathematics course you should take depends on your current mathematics skill level
Which Mathematics Course Should You Take? August, 018 Which mathematics course you should take depends on your current mathematics skill level and your intended major. This is a conversation you should
More informationPreCalculus. American Heritage Upper School Summer Math Packet
! PreCalculus American Heritage Upper School Summer Math Packet All Upper School American Heritage math students are required to complete a summer math packet. This packet is intended for all students
More informationPrecalculus Notes: Unit P Prerequisite Skills
Syllabus Objective Note: Because this unit contains all prerequisite skills that were taught in courses prior to precalculus, there will not be any syllabus objectives listed. Teaching this unit within
More informationEssential Question How can you cube a binomial? Work with a partner. Find each product. Show your steps. = (x + 1) Multiply second power.
4.2 Adding, Subtracting, and Multiplying Polynomials COMMON CORE Learning Standards HSA-APR.A.1 HSA-APR.C.4 HSA-APR.C.5 Essential Question How can you cube a binomial? Cubing Binomials Work with a partner.
More informationSimplifying Rational Expressions
.3 Simplifying Rational Epressions What are the ecluded values of a rational epression? How can you simplify a rational epression? ACTIVITY: Simplifying a Rational Epression Work with a partner. Sample:
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 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 informationSECTION P.5. Factoring Polynomials. Objectives. Critical Thinking Exercises. Technology Exercises
BLITMCPB.QXP.0599_48-74 2/0/02 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises 98. The common cold is caused by a rhinovirus. The polynomial -0.75 4 + + 5
More informationSection 5.5 Complex Numbers
Name: Period: Section 5.5 Comple Numbers Objective(s): Perform operations with comple numbers. Essential Question: Tell whether the statement is true or false, and justify your answer. Every comple number
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 informationLesson 10.1 Polynomials
Lesson 10.1 Polynomials Objectives Classify polynomials. Use algebra tiles to add polynomials. Add and subtract polynomials. A contractor is buying paint to cover the interior of two cubical storage tanks.
More informationVisit us at: for a wealth of information about college mathematics placement testing!
North Carolina Early Mathematics Placement Testing Program, 9--4. Multiply: A. 9 B. C. 9 9 9 D. 9 E. 9 Solution and Answer to Question # will be provided net Monday, 9-8-4 North Carolina Early Mathematics
More informationNote-Taking Guides. How to use these documents for success
1 Note-Taking Guides How to use these documents for success Print all the pages for the module. Open the first lesson on the computer. Fill in the guide as you read. Do the practice problems on notebook
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Math 2 Stud Guide-Chapters 8 and 9 Name Date: Time: MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all square roots of the number. ) 600 9,
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 informationINTRODUCTION GOOD LUCK!
INTRODUCTION The Summer Skills Assignment for has been developed to provide all learners of our St. Mar s Count Public Schools communit an opportunit to shore up their prerequisite mathematical skills
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 informationand Transitional Comprehensive Curriculum. Algebra II Unit 4: Radicals and the Complex Number System
01-1 and 01-14 Transitional Comprehensive Curriculum Algebra II Unit 4: Radicals and the Complex Number System Time Frame: Approximately three weeks Unit Description This unit expands student understanding
More informationSolving Equations with Variables on Both Sides
1. Solving Equations with Variables on Both Sides Essential Question How can you solve an equation that has variables on both sides? Perimeter Work with a partner. The two polygons have the same perimeter.
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 informationCourse Text. Course Description. Course Objectives. Course Prerequisites. Important Terms. StraighterLine Introductory Algebra
Introductory Algebra Course Text Dugopolski, Mark. Elementary Algebra, 6th edition. McGraw-Hill, 2009. ISBN 9780077224790 [This text is available as an etextbook at purchase or students may find used,
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS ALGEBRA II. 1 st Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS ALGEBRA II 1 st Nine Weeks, 2016-2017 OVERVIEW Algebra II Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
More informationGearing Up for Geometry!
Gearing Up for Geometry! Geometry is right around the corner and you need to make sure you are ready! Many of the concepts you learned in Algebra I will be used in Geometry and you will be epected to remember
More informationDivisibility Rules Algebra 9.0
Name Period Divisibility Rules Algebra 9.0 A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following eercise: 1. Cross
More informationMath 20-1 Functions and Equations Multiple Choice Questions
Math 0-1 Functions and Equations Multiple Choice Questions 1 7 18 simplifies to: A. 9 B. 10 C. 90 D. 4 ( x)(4 x) simplifies to: A. 1 x B. 1x 1 4 C. 1x D. 1 x 18 4 simplifies to: 6 A. 9 B. 4 C. D. 7 4 The
More informationMy Math Plan Assessment #1 Study Guide
My Math Plan Assessment #1 Study Guide 1. Find the x-intercept and the y-intercept of the linear equation. 8x y = 4. Use factoring to solve the quadratic equation. x + 9x + 1 = 17. Find the difference.
More informationLake Elsinore Unified School District Pacing Guide & Benchmark Assessment Schedule Algebra 1 Essentials
1.0 Students identify and use the arithmetic properties of subsets of integers, including closure properties for the four basic arithmetic operations where applicable: 1.1 Students use properties of numbers
More information7.2 Multiplying Polynomials
Locker LESSON 7. Multiplying Polynomials Teas Math Standards The student is epected to: A.7.B Add, subtract, and multiply polynomials. Mathematical Processes A.1.E Create and use representations to organize,
More informationMath 0210 Common Final Review Questions (2 5 i)(2 5 i )
Math 0 Common Final Review Questions In problems 1 6, perform the indicated operations and simplif if necessar. 1. ( 8)(4) ( )(9) 4 7 4 6( ). 18 6 8. ( i) ( 1 4 i ) 4. (8 i ). ( 9 i)( 7 i) 6. ( i)( i )
More information12.2 Simplifying Radical Expressions
x n a a m 1 1 1 1 Locker LESSON 1. Simplifying Radical Expressions Texas Math Standards The student is expected to: A.7.G Rewrite radical expressions that contain variables to equivalent forms. Mathematical
More informationRadical and. Exponential Functions
Preview of Algebra II Radical and Exponential Functions 11A Radical Functions and Equations 11-1 Square-Root Functions 11-2 Radical Expressions 11-3 Adding and Subtracting Radical Expressions 11-4 Multiplying
More informationSolve Quadratic Equations by Completing the Square
10.5 Solve Quadratic Equations by Completing the Square Before You solved quadratic equations by finding square roots. Now You will solve quadratic equations by completing the square. Why? So you can solve
More informationNFC ACADEMY COURSE OVERVIEW
NFC ACADEMY COURSE OVERVIEW Algebra I Fundamentals is a full year, high school credit course that is intended for the student who has successfully mastered the core algebraic concepts covered in the prerequisite
More informationREVIEW KEY VOCABULARY REVIEW EXAMPLES AND EXERCISES
Etra Eample. Graph.. 6. 7. (, ) (, ) REVIEW KEY VOCABULARY quadratic function, p. 6 standard form of a quadratic function, p. 6 parabola, p. 6 verte, p. 6 ais of smmetr, p. 6 minimum, maimum value, p.
More informationMath Analysis Chapter 2 Notes: Polynomial and Rational Functions
Math Analysis Chapter Notes: Polynomial and Rational Functions Day 13: Section -1 Comple Numbers; Sections - Quadratic Functions -1: Comple Numbers After completing section -1 you should be able to do
More informationDefinition: Quadratic equation: A quadratic equation is an equation that could be written in the form ax 2 + bx + c = 0 where a is not zero.
We will see many ways to solve these familiar equations. College algebra Class notes Solving Quadratic Equations: Factoring, Square Root Method, Completing the Square, and the Quadratic Formula (section
More informationMathematics GRADE 8 Teacher Packet
COMMON CORE Standards Plus Mathematics GRADE 8 Teacher Packet Copyright 01 Learning Plus Associates All Rights Reserved; International Copyright Secured. Permission is hereby granted to teachers to reprint
More informationSolutions of Linear Equations
Lesson 14 Part 1: Introduction Solutions of Linear Equations Develop Skills and Strategies CCSS 8.EE.C.7a You ve learned how to solve linear equations and how to check your solution. In this lesson, you
More informationAre You Ready? Find Area in the Coordinate Plane
SKILL 38 Are You Read? Find Area in the Coordinate Plane Teaching Skill 38 Objective Find the areas of figures in the coordinate plane. Review with students the definition of area. Ask: Is the definition
More informationCommon Core State Standards for Activity 14. Lesson Postal Service Lesson 14-1 Polynomials PLAN TEACH
Postal Service Lesson 1-1 Polynomials Learning Targets: Write a third-degree equation that represents a real-world situation. Graph a portion of this equation and evaluate the meaning of a relative maimum.
More informationOne of your primary goals in mathematics should be to become a good problem solver. It helps to approach a problem with a plan.
PROBLEM SOLVING One of our primar goals in mathematics should be to become a good problem solver. It helps to approach a problem with a plan. Step Step Step Step Understand the problem. Read the problem
More informationDay 3: Section P-6 Rational Expressions; Section P-7 Equations. Rational Expressions
1 Day : Section P-6 Rational Epressions; Section P-7 Equations Rational Epressions A rational epression (Fractions) is the quotient of two polynomials. The set of real numbers for which an algebraic epression
More informationGRE Quantitative Reasoning Practice Questions
GRE Quantitative Reasoning Practice Questions y O x 7. The figure above shows the graph of the function f in the xy-plane. What is the value of f (f( ))? A B C 0 D E Explanation Note that to find f (f(
More information1-2 Study Guide and Intervention
1- Study Guide and Intervention Real Numbers All real numbers can be classified as either rational or irrational. The set of rational numbers includes several subsets: natural numbers, whole numbers, and
More informationChapter 4 Polynomial and Rational Functions
Chapter Polynomial and Rational Functions - Polynomial Functions Pages 09 0 Check for Understanding. A zero is the value of the variable for which a polynomial function in one variable equals zero. A root
More information