CHAPTER 8 ROOTS AND RADICALS
|
|
- Janel Garrison
- 7 years ago
- Views:
Transcription
1 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 Find the square root , Objective Decide whether a given root is rational, irrational, or not a real number. Tell whether the square root is rational, irrational, or not a real number
2 10 Chapter Eight Additional Exercises Objective Find decimal approximations for irrational square roots. Use a calculator to find a decimal approximation for the root. Round answers to the nearest thousandth , 00 8., 8 Objective Use the Pythagorean formula. Find the length of the unknown side of the right triangle with sides a, b, and c, where c is the hypotenuse. If necessary, round your answer to the nearest thousandth. 9. a = 8, b = 1 0. a = 8, b = 1 1. c =, a =. c =, b =. c = 1, a = 1. a =, b = 8 Use the Pythagorean formula to solve the problem. If necessary, round your answer to the nearest thousandth.. The hypotenuse of a right triangle measure 10 centimeters and one leg measures centimeters. How long is the other leg?. Two sides of a rectangle measure centimeters and 11 centimeters. How long are the diagonals of the rectangle?. A diagonal of a rectangle measures 9 inches. The length of the rectangle is 1 inches. Find the width of the rectangle. 8. A diagonal of a rectangle measures. meters. The width of the rectangle is 1. meters. Find the length of the rectangle.
3 Chapter Eight Additional Exercises A ladder feet long leans against a wall. The foot of the ladder is feet from the base of the wall. How high on the wall does the top of the ladder rest? 0. Laura is flying a kite on 0 feet of string. How high is the kite above her hand (vertically) if the horizontal distance between Laura and the kite is feet? 1. Kevin started to drive due south at the same time Lydia started to drive due west. Lydia drove 1 miles in the same time that Kevin drove 8 miles. How far apart were they at that time?. A cable from the top of a pole 1 feet tall is pulled taut and attached to the ground 8 feet from the base of the pole. How long is the cable?
4 1 Chapter Eight Additional Exercises. The foot of a loading ramp 9 feet long is placed feet from the base of a platform. The top of the ramp rests on the platform. How high is the platform?. A plane flies due east for miles and then due south until it is miles from its starting point. How far south did the plane fly? Objective Use the distance formula. Find the distance between each pair of points.. (,),(,8). (,),( 9,1). (,),( 0, ) 8. (, ),(, ) 9. (, 1 ),(, ) 0. ( 0,0),(, ) 1. (, ),(, ). (,0),( 0, ). (, ),(,1). (, 8),(,1). (, ),(, ). (, 11 ),( 9, ) Objective Find cube, fourth, and other roots. Find each root that is a real number ,
5 Chapter Eight Additional Exercises 1 Mixed Exercises Find each root that is a real number Write rational, irrational, or not a real number to describe the number. If the number is rational, give its exact value. If the number is irrational, give a decimal approximation to the nearest thousandth. Use a calculator as necessary Find the length of the unknown side of the right triangle with sides a, b, and c, where c is the hypotenuse. If necessary, round your answer to the nearest thousandth a = 10, b = 110. b =, c = a = 1, c = c = 1, a = a =, b = b = 8, c = a = 9, c = b =, c = 11 Writing/Conceptual Exercises Answer the question. 11. How many fourth roots does 0 have? 118. How many real number fourth roots does any positive number have? 119. How many real number fourth roots does any negative number have?
6 1 Chapter Eight Additional Exercises 10. Which of the following numbers have rational square roots? (a) (b) 9 (c) 9 (d) Which of the following numbers have irrational square roots? (a) (b) 1 (c) 1 (d) 9 What must be true about b for the statement to be true? 1. b is a negative number. 1. b is a negative number. 1. b is a positive number. 1. b is a positive number. 1. b is zero. Section 8. Multiplying, Dividing, and Simplifying Radicals Objective 1 Multiply square root radicals. Use the product rule for radicals to find the product r, r > 0 1. x, x > a b, a > 0, b > 0
7 Chapter Eight Additional Exercises 1 Objective Simplify radicals by using the product rule. Simplify the radical. Assume that all variables represent nonnegative real numbers y 18. p q a 10. 1r s Find the product and simplify Objective Simplify radicals by using the quotient rule. Use the quotient rule and product rule, as necessary, to simplify the expression Objective Simplify radicals involving variables x 1. p 18. x y 19. a 0a x 1. 81x 1. x 8 1. x
8 1 Chapter Eight Additional Exercises Objective Simplify higher roots. Simplify the expression Mixed Exercises Find the product and simplify if possible Simplify the expression. Assume that all variables represent positive numbers x x a 0. a a b y x 10.
9 Chapter Eight Additional Exercises 1 Writing/Conceptual Exercises Decide whether the statement is true or false. 11. = 1. + = + 1. ( 10) = ( 10) = Which of the following radicals are simplified according to the guidelines given in the textbook? (a) (b) 19 (c) 1 (d) 1. Which of the following radicals are simplified according to the guidelines given in the textbook? (a) 9 (b) 1 (c) 1 (d) 0 Section 8. Adding and Subtracting Radicals Objective 1 Add and subtract radicals. Add or subtract, as indicated Objective Simplify radical sums and differences. Simplify and add or subtract terms wherever possible. Assume that all variables represent nonnegative real numbers
10 18 Chapter Eight Additional Exercises z 18z 8. 9w w 9. 0x 00x r + r Objective Simplify more complicated radical expressions. Perform the indicated operations. Assume that all variables represent nonnegative real numbers x + x 8. y 18 y 9. 10x 1x x w 0w 8w 1. k k + k k. y 80y Mixed Exercises Simplify and perform the indicated operations. Assume that all variables represent nonnegative real numbers m m 8. 1
11 Chapter Eight Additional Exercises m 0. r 0 8r z + z + z y + 1y + y. 1r + r 1r x 9x + x Writing/Conceptual Exercises 9. Write an equation showing how the distributive property is used to justify the statement + 8 = Write an equation showing how the distributive property is used to justify the statement 11 =. Despite the fact that and 8 are radicals that have different root indexes, they can be added to obtain a single term: + 8 = + =. 81. Make up a similar sum of radicals that leads to an answer of Make up a similar difference of radicals that leads to an answer of 8.
12 0 Chapter Eight Additional Exercises Section 8. Rationalizing the Denominator Objective 1 Rationalize denominators with square roots. Rationalize the denominator. Write all answers in simplest form Objective Write radicals in simplified form. Perform the indicated operations and write all answers in simplest form. Rationalize all denominators. Assume that all variables represent positive real numbers y x 09. r s 10. a b 11. k k m df 1. d 1. 0a b a
13 Chapter Eight Additional Exercises 1 Objective Rationalize denominators with cube roots. Rationalize the denominator. Assume that all variables in the denominator represent nonzero real numbers r s. 9 r. v w 8t 8. u Mixed Exercises Simplify the expression. Assume that all variables represent positive real numbers x 8. 1 qt t 0. x 81y 1. 1 r
14 Chapter Eight Additional Exercises. 1 t x.. y t 1x. t x c d. t Writing/Conceptual Exercises. Which one of the following would be an appropriate choice for multiplying the numerator and denominator of in order to rationalize the denominator? (a) (b) (c) 9 (d). Which one of the following would be an appropriate choice for multiplying the y numerator and denominator of in order to rationalize the denominator? 10z (a) 10z (b) 100z (c) 10z (d) 100z. Which one of the following would be an appropriate choice for multiplying the s numerator and denominator of in order to rationalize the denominator? 9r (a) r (b) 9r (c) r (d) s. What would be your first step in simplifying the radical? What property would 11 you be using in this step?
15 Chapter Eight Additional Exercises Section 8. More Simplifying and Operations with Radicals Objective 1 Simplify products of radical expressions. Simplify the expression.. ( ) ( + ) 9. ( 1 + ) 0. ( 11 ) 1. ( + )( + ). ( 11)( + ). ( )( + 10). ( )( + ). ( + 10)( 10). ( )( + ). ( ) 8. ( 10 11) 9. ( + ) 0. ( + ) Objective Use conjugates to rationalize denominators of radical expressions. Rationalize the denominator
16 Chapter Eight Additional Exercises Objective Write radical expressions with quotients in lowest terms. Write the quotient in lowest terms Mixed Exercises Simplify the expression. 9. ( 11) 9. ( )( + 1) 9. ( 11 + ) 98. ( + 1)( 1) ( 1 + )( 1 ) 0. ( + ) 0. ( 1) 08. ( 10 + )( 11) 09. ( )( )
17 Chapter Eight Additional Exercises Write the quotient in lowest terms Section 8. Solving Equations with Radicals Objective 1 Solve radical equations having square root radicals. Solve the equation. 1. w = 9. q 1 = 0. 8 y =. b +1 =. q + =. z = 0. + k = k 8. r + 1 = r 9. x + = x 0. t + = 0 1. t + = t 1. x + 18 = x + 1
18 Chapter Eight Additional Exercises Objective Identify equations with no solutions. Solve the equation if it has a solution.. y = p = 0. z + = 0. 0 = r. y + = 1 8. k + = 0 9. x + + = 0 0. m + 1 = m + 1. n = n. p = p +. r = r r + 1. s = s + s +. b = b b + 1. d + d d = 0 Objective Solve equations by squaring a binomial. Find all solutions for the equation.. y + 1 = y m + = m + 9. x + = x 0. x = x 1. t + = t + t +. q 1 = q + 8q p = p +. a 1 = a + 1. x = x 8. y + 1 = y + 1. b = b 8. p + 1 = p c + = c + 0. t = t + 1. x + 1 = x + 9. x x + = 0
19 Chapter Eight Additional Exercises. t + t = 1. a = a +. k + = k + 1. x + 1 = + x. x + = x r + r + 11 = Objective Solve radical equations having cube root radicals. Solve each equation. 9. x+ 1= x 0. x = x+ 1. x = x. x = x. x + 1x 8 = x. 8x + x = x Mixed Exercises Find all solutions for the equation.. z 8 =. q + =. r + = 0 8. p + = 0 9. r = 8r y = y x+ 1= x x + 1 = x 8. n + = n 8. m + = 8. t = t t+ 8. t = 9t t x + = x 88. m + = m p = p p k k k = + 10
20 8 Chapter Eight Additional Exercises 91. x = x r = r + r 0 9. x + 1 = x 1 9. x + 1 = x y = 0 9. x = x 9. y + 1 = y a + a + = 99. x 1 + = c + = c q 8 = q 0. k + 10 = k x+ = x 0. x = x 0. x = x 0. x = x 9 0. x + x = 10x 08. x + x = 1x Writing/Conceptual Exercises 09. How can you tell that the equation x + 1 = 9 has no real number solution without performing any algebraic steps? 10. Explain why the equation x = 81 has two real number solutions, while the equation x = 9 has only one real number solution. 11. A student is told that he must check his solutions to the equation r = r. The student said that he doesn t see why this is necessary because he is sure he didn t make any mistakes in solving the equation. How would you respond? 1. Explain why the equation z = z + z + cannot have a negative solution.
21 Chapter Eight Additional Exercises 9 Section 8. Using Rational Numbers as Exponents Objective 1 Define and use expressions of the form Evaluate the expression. n a 1 /. 1. 1/ / 1. 1/ / 1. 1/ / 19. 1/ 0. 1/ / 1. 1/ 1. 1/ 9. 1/ / 1. 1/ 81. 1/ 8. 1/ / 0. 1/ 1. 1/. 1/ 18 Objective Define and use expressions of the form m n a /. Evaluate the expression.. / 1. /. /. /. / 8 8. / 8 9. / 0. / 1. /. / 1. / 1. / 1. / 8. /. / 8. / 9. / 1 0. / 1. / / Objective Apply the rules for exponents using rational exponents. Simplify the expression. Write the answer in exponential form with only positive exponents. Assume that all variables represent positive numbers.. 1/ /. / /. / 8 / 8. 8 / /. 8 / 8 / / 9 / / 9. 1/ 8 1/ / / / 8
22 0 Chapter Eight Additional Exercises. ( r s / ) /. ( 9 / ). ( / ) /.. 1/ 1 9 /. / 8. a b / 1/ 1 9. r s 1/ 1/ / 1/ y y 0. y / 1. r r r / 1/ / / / z z. / z x. x 1/ / /. c x / 1 1/. ( ) 1/ ( ) a a Objective Use rational exponents to simplify radicals. Simplify the radical by first writing it in exponential form. Give the answer as an integer or a radical in simplest form. Assume that all variables represent nonnegative numbers p 8. z 8. r 8. 8 a Mixed Exercises Evaluate the expression / 90. / ( / ) / 9. / 9. 1/ / /
23 Chapter Eight Additional Exercises / / 01. 1/ 100, / 0. / / 9 / / Simplify the expression. Write the answer in exponential form with only positive exponents. Assume that all variables represent positive numbers. y 0. 1/ / y y / x x 08. / x / 09. x y / / 10. / 8 1 1/ ( / / ) 1/ r s 1. a b 1/ 1/ 1/ Writing/Conceptual Exercises Decide which one of the four choices is not equal to the given expression. 1. 1/ 11 (a). 11 (b) 11 (c) 11 (d) / (a) (b) (c) 10 (d) / 1000 (a) 10 (b) 1 10 (c) (d) / 8 (a) 1 (b) 8 (c) 1 1 (d)
24
12) 13) 14) (5x)2/3. 16) x5/8 x3/8. 19) (r1/7 s1/7) 2
DMA 080 WORKSHEET # (8.-8.2) Name Find the square root. Assume that all variables represent positive real numbers. ) 6 2) 8 / 2) 9x8 ) -00 ) 8 27 2/ Use a calculator to approximate the square root to decimal
More informationFree Pre-Algebra Lesson 55! page 1
Free Pre-Algebra Lesson 55! page 1 Lesson 55 Perimeter Problems with Related Variables Take your skill at word problems to a new level in this section. All the problems are the same type, so that you can
More informationMultiplying and Dividing Radicals
9.4 Multiplying and Dividing Radicals 9.4 OBJECTIVES 1. Multiply and divide expressions involving numeric radicals 2. Multiply and divide expressions involving algebraic radicals In Section 9.2 we stated
More informationApplications of the Pythagorean Theorem
9.5 Applications of the Pythagorean Theorem 9.5 OBJECTIVE 1. Apply the Pythagorean theorem in solving problems Perhaps the most famous theorem in all of mathematics is the Pythagorean theorem. The theorem
More informationMultiplication and Division Properties of Radicals. b 1. 2. a Division property of radicals. 1 n ab 1ab2 1 n a 1 n b 1 n 1 n a 1 n b
488 Chapter 7 Radicals and Complex Numbers Objectives 1. Multiplication and Division Properties of Radicals 2. Simplifying Radicals by Using the Multiplication Property of Radicals 3. Simplifying Radicals
More informationHow do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.
The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics
More informationSOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root 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 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 informationPythagorean Theorem: 9. x 2 2
Geometry Chapter 8 - Right Triangles.7 Notes on Right s Given: any 3 sides of a Prove: the is acute, obtuse, or right (hint: use the converse of Pythagorean Theorem) If the (longest side) 2 > (side) 2
More informationQuick Reference ebook
This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed
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 informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
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 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 informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte
More informationBig Bend Community College. Beginning Algebra MPC 095. Lab Notebook
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
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 informationSECTION 1-6 Quadratic Equations and Applications
58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
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 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 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 informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationof surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433
Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}
More informationCOWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2
COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level This study guide is for students trying to test into College Algebra. There are three levels of math study guides. 1. If x and y 1, what
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationNonlinear Systems and the Conic Sections
C H A P T E R 11 Nonlinear Systems and the Conic Sections x y 0 40 Width of boom carpet Most intense sonic boom is between these lines t a cruising speed of 1,40 miles per hour, the Concorde can fly from
More information7.2 Quadratic Equations
476 CHAPTER 7 Graphs, Equations, and Inequalities 7. Quadratic Equations Now Work the Are You Prepared? problems on page 48. OBJECTIVES 1 Solve Quadratic Equations by Factoring (p. 476) Solve Quadratic
More informationSIMPLIFYING SQUARE ROOTS
40 (8-8) 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
More informationGeometry and Measurement
The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for
More informationSquare Roots and the Pythagorean Theorem
4.8 Square Roots and the Pythagorean Theorem 4.8 OBJECTIVES 1. Find the square root of a perfect square 2. Use the Pythagorean theorem to find the length of a missing side of a right triangle 3. Approximate
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
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 informationMTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 17, 2006
MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions Created January 7, 2006 Math 092, Elementary Algebra, covers the mathematical content listed below. In order
More informationMTH 100 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created June 6, 2011
MTH 00 College Algebra Essex County College Division of Mathematics Sample Review Questions Created June 6, 0 Math 00, Introductory College Mathematics, covers the mathematical content listed below. In
More informationModuMath Basic Math Basic Math 1.1 - Naming Whole Numbers Basic Math 1.2 - The Number Line Basic Math 1.3 - Addition of Whole Numbers, Part I
ModuMath Basic Math Basic Math 1.1 - Naming Whole Numbers 1) Read whole numbers. 2) Write whole numbers in words. 3) Change whole numbers stated in words into decimal numeral form. 4) Write numerals in
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
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 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 informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
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 informationSolve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers
More informationSQUARE-SQUARE ROOT AND CUBE-CUBE ROOT
UNIT 3 SQUAREQUARE AND CUBEUBE (A) Main Concepts and Results A natural number is called a perfect square if it is the square of some natural number. i.e., if m = n 2, then m is a perfect square where m
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationGeometry: Classifying, Identifying, and Constructing Triangles
Geometry: Classifying, Identifying, and Constructing Triangles Lesson Objectives Teacher's Notes Lesson Notes 1) Identify acute, right, and obtuse triangles. 2) Identify scalene, isosceles, equilateral
More informationPERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.
PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the
More informationVocabulary Cards and Word Walls Revised: June 29, 2011
Vocabulary Cards and Word Walls Revised: June 29, 2011 Important Notes for Teachers: The vocabulary cards in this file match the Common Core, the math curriculum adopted by the Utah State Board of Education,
More informationA.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it
Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply
More informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More informationMath 0306 Final Exam Review
Math 006 Final Exam Review Problem Section Answers Whole Numbers 1. According to the 1990 census, the population of Nebraska is 1,8,8, the population of Nevada is 1,01,8, the population of New Hampshire
More informationMATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab
MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will
More informationTo Evaluate an Algebraic Expression
1.5 Evaluating Algebraic Expressions 1.5 OBJECTIVES 1. Evaluate algebraic expressions given any signed number value for the variables 2. Use a calculator to evaluate algebraic expressions 3. Find the sum
More informationPrinciples of Mathematics MPM1D
Principles of Mathematics MPM1D Grade 9 Academic Mathematics Version A MPM1D Principles of Mathematics Introduction Grade 9 Mathematics (Academic) Welcome to the Grade 9 Principals of Mathematics, MPM
More informationScope and Sequence KA KB 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B
Scope and Sequence Earlybird Kindergarten, Standards Edition Primary Mathematics, Standards Edition Copyright 2008 [SingaporeMath.com Inc.] The check mark indicates where the topic is first introduced
More informationSect 6.7 - Solving Equations Using the Zero Product Rule
Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred
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 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 informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
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 informationMcDougal Littell California:
McDougal Littell California: Pre-Algebra Algebra 1 correlated to the California Math Content s Grades 7 8 McDougal Littell California Pre-Algebra Components: Pupil Edition (PE), Teacher s Edition (TE),
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationExpression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds
Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative
More informationMATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationSummer Math Exercises. For students who are entering. Pre-Calculus
Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn
More informationWeek 13 Trigonometric Form of Complex Numbers
Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working
More informationFSCJ PERT. Florida State College at Jacksonville. assessment. and Certification Centers
FSCJ Florida State College at Jacksonville Assessment and Certification Centers PERT Postsecondary Education Readiness Test Study Guide for Mathematics Note: Pages through are a basic review. Pages forward
More informationCOMPETENCY TEST SAMPLE TEST. A scientific, non-graphing calculator is required for this test. C = pd or. A = pr 2. A = 1 2 bh
BASIC MATHEMATICS COMPETENCY TEST SAMPLE TEST 2004 A scientific, non-graphing calculator is required for this test. The following formulas may be used on this test: Circumference of a circle: C = pd or
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More information8-6 Radical Expressions and Rational Exponents. Warm Up Lesson Presentation Lesson Quiz
8-6 Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt Algebra ALgebra2 2 Warm Up Simplify each expression. 1. 7 3 7 2 16,807 2. 11 8 11 6 121 3. (3 2 ) 3 729 4. 5.
More informationFlorida Math 0018. Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies - Lower
Florida Math 0018 Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies - Lower Whole Numbers MDECL1: Perform operations on whole numbers (with applications, including
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationAlgebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test
Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action
More informationMATH 100 PRACTICE FINAL EXAM
MATH 100 PRACTICE FINAL EXAM Lecture Version Name: ID Number: Instructor: Section: Do not open this booklet until told to do so! On the separate answer sheet, fill in your name and identification number
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More information43 Perimeter and Area
43 Perimeter and Area Perimeters of figures are encountered in real life situations. For example, one might want to know what length of fence will enclose a rectangular field. In this section we will study
More information13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant
æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the
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 informationBasic Lesson: Pythagorean Theorem
Basic Lesson: Pythagorean Theorem Basic skill One leg of a triangle is 10 cm and other leg is of 24 cm. Find out the hypotenuse? Here we have AB = 10 and BC = 24 Using the Pythagorean Theorem AC 2 = AB
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.
More informationSuch As Statements, Kindergarten Grade 8
Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential
More informationAlgebra 1 Course Information
Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationThe GED math test gives you a page of math formulas that
Math Smart 643 The GED Math Formulas The GED math test gives you a page of math formulas that you can use on the test, but just seeing the formulas doesn t do you any good. The important thing is understanding
More informationGeometry Notes RIGHT TRIANGLE TRIGONOMETRY
Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right
More informationCAMI Education linked to CAPS: Mathematics
- 1 - TOPIC 1.1 Whole numbers _CAPS curriculum TERM 1 CONTENT Mental calculations Revise: Multiplication of whole numbers to at least 12 12 Ordering and comparing whole numbers Revise prime numbers to
More informationVOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region.
Math 6 NOTES 7.5 Name VOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region. **The formula for the volume of a rectangular prism is:** l = length w = width h = height Study Tip:
More informationVoyager Sopris Learning Vmath, Levels C-I, correlated to the South Carolina College- and Career-Ready Standards for Mathematics, Grades 2-8
Page 1 of 35 VMath, Level C Grade 2 Mathematical Process Standards 1. Make sense of problems and persevere in solving them. Module 3: Lesson 4: 156-159 Module 4: Lesson 7: 220-223 2. Reason both contextually
More information6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More information9 Right Triangle Trigonometry
www.ck12.org CHAPTER 9 Right Triangle Trigonometry Chapter Outline 9.1 THE PYTHAGOREAN THEOREM 9.2 CONVERSE OF THE PYTHAGOREAN THEOREM 9.3 USING SIMILAR RIGHT TRIANGLES 9.4 SPECIAL RIGHT TRIANGLES 9.5
More information