Sect Definitions of a 0 and a n

Size: px
Start display at page:

Download "Sect Definitions of a 0 and a n"

Transcription

1 5 Sect 5. - Definitions of a 0 and a n Concept # Definition of a 0. Let s examine the quotient rule when the powers are equal. Simplify: Ex. 5 5 There are two ways to view this problem. First, any non-zero number divided by itself is, so, 5. But, using the quotient 5 rule, This says that 0. We can do this same trick with any base except for zero. Zero Exponents If a is any non-zero real number, then a 0. Simplify: Ex. a 0 Ex. b ( ) 0 Ex. c 0 Ex. d x 0 Ex. e x y 0 Ex. f (x y) 0 Ex. g (x y) 0 Ex. h ( x y) 0 a) 0 b) ( ) 0 c) 0 () (the 0 exponent only applies to ) d) x 0 () (the 0 exponent only applies to x) e) x y 0 x () x. (the 0 exponent only applies to y) f) (x y) 0 (). (the 0 exponent only applies to x y) g) (x y) 0 (). (the 0 exponent only applies to x y) h) ( x y) 0 (). Concept # Definition of a n Let s examine the quotient rule when the power in the denominator is larger than the power in the numerator.

2 5 Simplify: Ex. 5 8 There are two ways to view this problem. First, 5 and 8 56, then which reduces to 8. But, 8 5,so,. 8 But, using the quotient rule, This says that. We can do this same trick with any base except for zero. Also, since Negative Exponents If a and b are any non-zero real numbers, then a n and a n b n bn a. Note this also implies that ( ) n a n b b n b ( ) n. b n a n a In words, when raising a quantity to a negative power, take the reciprocal of the base and change the sign of the exponent. Simplify the following. Write your answer using positive exponents: Ex. 4a Ex. 4b ( ) 4 Ex. 4c Ex. 4e 5 Ex. 4d 5 ( ) Ex. 4f ( x ) Ex. 4g ( 4 y ) Ex. 4h ( ) 8 a).

3 b) ( ) 4 ( ) c) d) 5 ( ) 4 e) 5 ( ) 4 (exponents) 5 6 (multiplication) f) ( x ) ( x ) ( ) (x ) ( ) x 6. 7x 6 (#4 power of a product rule) (# power rule) g) ( 4 y ) ( y 4 ) (#5 power of a quotient rule) y 4 y 64.

4 54 h) ( ) 8 (8) 64 Ex. 5a 5x Ex. 5b (5x) 4x Ex. 5c y Ex. 5d ( ) a b 7 c 5 5a b 5 ( 7) q 4 r v 0 a) 5x (write over ) 5x 5 x. b) (5x) (write over ) (5x) (5x). 5x (#4 power of a product rule and simplify) c) If the exponents are already positive, do not move the factors. Only move the factors that have negative exponents: 4x y 5a b ` 4x y 5 (negative positive is negative) 5a b 5 4b5 y 5a x. so we do not move it). (Note 4 is not an exponent, but a number d) If the exponents are already positive, do not move the factors. Only move the factors that have negative exponents: ( ) a b 7 c 5 ( 7) q 4 r v 0 8a b 7 c 5 (negative 49q 4 r 0 positive is negative) v

5 55 Concept # 8a b 7 c 5 49q 4 r v 0 8b 7 q 4 49a c 5 r v 0 But v 0, so 8b 7 q 4 8b 7 q 4 49a c 5 r v 0 49a c 5 r () 8b7 q4 49a c 5 r Properties of Integral Exponents: A Summary We can now extend the properties of exponents discussed in sections 5. and 5. to include integral exponents. Properties of Integral Exponents Assume that a and b are non-zero real numbers and m and n are integers. Property Example Notes The Product Rule. a m a n a m + n x x 5 x + 5 x 8 x x 5 (x x x)(x x x x x) x 8 The Quotient Rule. b m b n bm n b The Power Rule 6 b b6 b 4 b 6 b b b b b b b b b b 4. (a m ) n a m n (x ) x x 6 (x ) (x x x)(x x x) x 6 Power of a Product Rule 4. (ab) n a n b n (ab) 4 a 4 b 4 (ab) 4 (ab)(ab)(ab)(ab) a 4 b 4 Power of a Quotient Rule 5. ( a ) n an a ( ) 4 a ( a4 b b n b b 4 b ) 4 ( a a a a )( )( )( ) a4 b b b b b 4 Definitions Assume that a is non-zero real number and n is an integers. Definition Example Notes Zero Exponents a 0 ( 5) 0 Negative Exponents a n x 7 a n x 7 Any non-zero real number raised to the 0 power is. Take the reciprocal of the base and change the sign of the exponent.

6 Concept #4 Simplifying Expressions with Exponents Simplify the following. Write your answer using positive exponents: Ex. 6a x h 7 v 8 h 9 Ex. 6b ( 7a c 4 ) Ex. 6c ( 5x y xy 4 ) a) x h 7 v 8 h 9 x h v 8 v8 x h. Ex. 6d ( 7x y 4 ) (x y ) (# product rule in the denominator) b) ( 7a c 4 ) (#4 power of a product rule) ( 7) (a ) (c 4 ) (# power rule) ( 7) a 6 c c ( 7) a 6 c 4a 6 56 c) ( 5x y xy 4 ) ( 5y x xy 4 ) ( 5y x y 4 ) ( 5y x ) ( 5 x y ) ( x y 5 ) (apply the definition of a negative exponent inside the parenthesis) (# product rule) (# quotient rule inside the parenthesis) (apply the definition of a negative exponent inside the parenthesis) (#5 power of a quotient rule)

7 57 d) x y (#4 power of a product rule) ( 5) (x ) (y) ( 5) x9 y ( 5) x9 y 5 ( 7x y 4 ) (x y ) (# power rule) (#4 power of a product rule) ( 7) (x ) (y 4 ) (# power rule) () (x ) (y ) ( 7) x 6 y 8 () x y x ( 7) x 6 y 8 y x ( 7) x 6 y 0 x 4 ( 7) y 0 ( 7) x 4 y 0 49x 4 y 0 (# product rule) (# quotient rule) Ex ( ) ( ) ( ) 7 0 (#5 quotient to a power and simplify) (LCD 47 and simplify)

Sect Least Common Denominator

Sect Least Common Denominator 4 Sect.3 - Least Common Denominator Concept #1 Writing Equivalent Rational Expressions Two fractions are equivalent if they are equal. In other words, they are equivalent if they both reduce to the same

More information

Sect Exponents: Multiplying and Dividing Common Bases

Sect Exponents: Multiplying and Dividing Common Bases 154 Sect 5.1 - Exponents: Multiplying and Dividing Common Bases Concept #1 Review of Exponential Notation In the exponential expression 4 5, 4 is called the base and 5 is called the exponent. This says

More information

Lesson 2. When the exponent is a positive integer, exponential notation is a concise way of writing the product of repeated factors.

Lesson 2. When the exponent is a positive integer, exponential notation is a concise way of writing the product of repeated factors. Review of Exponential Notation: Lesson 2 - read to the power of, where is the base and is the exponent - if no exponent is denoted, it is understood to be a power of 1 - if no coefficient is denoted, it

More information

Alg 1B Chapter 7 Final Exam Review

Alg 1B Chapter 7 Final Exam Review Name: Class: Date: ID: A Alg B Chapter 7 Final Exam Review Please answer all questions and show your work. Simplify ( 2) 4. 2. Simplify ( 4) 4. 3. Simplify 5 2. 4. Simplify 9x0 y 3 z 8. 5. Simplify 7w0

More information

Chapter 5: Exponents and Polynomials

Chapter 5: Exponents and Polynomials Chapter 5: Exponents and Polynomials 5.1 Multiplication with Exponents and Scientific Notation 5.2 Division with Exponents 5.3 Operations with Monomials 5.4 Addition and Subtraction of Polynomials 5.5

More information

Chapter 6. Polynomials

Chapter 6. Polynomials Chapter 6 Polynomials How to Play the Stock Market 6.1 Monomials: Multiplication and Division 6.2 Polynomials 6.3 Addition and Subtraction of Polynomials 6.4 Multiplication of Polynomials Chapter Review

More information

5.1. Integer Exponents and Scientific Notation. Objectives. Use the product rule for exponents. Define 0 and negative exponents.

5.1. Integer Exponents and Scientific Notation. Objectives. Use the product rule for exponents. Define 0 and negative exponents. Chapter 5 Section 5. Integer Exponents and Scientific Notation Objectives 2 4 5 6 Use the product rule for exponents. Define 0 and negative exponents. Use the quotient rule for exponents. Use the power

More information

Sect 2.4 Multiplying and Dividing Integers

Sect 2.4 Multiplying and Dividing Integers 55 Sect 2.4 Multiplying and Dividing Integers Objective a: Understanding how to multiply two integers. To see how multiplying and dividing a negative and a positive number works, let s look at some examples.

More information

Reteach Multiplying and Dividing Rational Expressions

Reteach Multiplying and Dividing Rational Expressions 8-2 Multiplying and Dividing Rational Expressions Examples of rational expressions: 3 x, x 1, and x 3 x 2 2 x 2 Undefined at x 0 Undefined at x 0 Undefined at x 2 When simplifying a rational expression:

More information

Sect Complex Numbers

Sect Complex Numbers 161 Sect 10.8 - Complex Numbers Concept #1 Imaginary Numbers In the beginning of this chapter, we saw that the was undefined in the real numbers since there is no real number whose square is equal to a

More information

INTRODUCTION TO FRACTIONS

INTRODUCTION TO FRACTIONS INTRODUCTION TO FRACTIONS MEANING AND PROPERTIES OF FRACTIONS Fractions are used to represent parts of a whole. Example: What is the fraction of the shaded area? one-half one-quarter three-eighths 4 The

More information

Prepared by Sa diyya Hendrickson. Package Summary

Prepared by Sa diyya Hendrickson. Package Summary Introduction Prepared by Sa diyya Hendrickson Name: Date: Figure 1: c Cengage Learning Package Summary Definition and Properties of Exponents Understanding Properties (Frayer Models) Discovering Zero and

More information

At the end of this section, you should be able to solve equations that are convertible to equations in linear or quadratic forms:

At the end of this section, you should be able to solve equations that are convertible to equations in linear or quadratic forms: Equations in Linear and Quadratic Forms At the end of this section, you should be able to solve equations that are convertible to equations in linear or quadratic forms: Equations involving rational expressions

More information

Name: Chapter 7: Exponents and Polynomials

Name: Chapter 7: Exponents and Polynomials Name: Chapter 7: Exponents and Polynomials 7-1: Integer Exponents Objectives: Evaluate expressions containing zero and integer exponents. Simplify expressions containing zero and integer exponents. You

More information

Section September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc.

Section September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc. Section 2.1-2.2 September 6, 2017 1 Polynomials Definition. A polynomial is an expression of the form a n x n + a n 1 x n 1 + + a 1 x + a 0 where each a 0, a 1,, a n are real numbers, a n 0, and n is a

More information

Sect Properties of Real Numbers and Simplifying Expressions

Sect Properties of Real Numbers and Simplifying Expressions Sect 1.7 - Properties of Real Numbers and Simplifying Expressions Concept #1 Commutative Properties of Real Numbers Ex. 1a 9.34 + 2.5 Ex. 1b 2.5 + ( 9.34) Ex. 1c 6.3(4.2) Ex. 1d 4.2( 6.3) a) 9.34 + 2.5

More information

Simplify each numerical expression. Show all work! Only use a calculator to check. 1) x ) 25 ( x 2 3) 3) 4)

Simplify each numerical expression. Show all work! Only use a calculator to check. 1) x ) 25 ( x 2 3) 3) 4) NAME HONORS ALGEBRA II REVIEW PACKET To maintain a high quality program, students entering Honors Algebra II are expected to remember the basics of the mathematics taught in their Algebra I course. In

More information

ADVANCED/HONORS ALGEBRA 2 - SUMMER PACKET

ADVANCED/HONORS ALGEBRA 2 - SUMMER PACKET NAME ADVANCED/HONORS ALGEBRA 2 - SUMMER PACKET Part I. Order of Operations (PEMDAS) Parenthesis and other grouping symbols. Exponential expressions. Multiplication & Division. Addition & Subtraction. Tutorial:

More information

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!! 1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a

More information

Chapter 3: Factors, Roots, and Powers

Chapter 3: Factors, Roots, and Powers Chapter 3: Factors, Roots, and Powers Section 3.1 Chapter 3: Factors, Roots, and Powers Section 3.1: Factors and Multiples of Whole Numbers Terminology: Prime Numbers: Any natural number that has exactly

More information

Math RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5

Math RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5 Math 201-203-RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5 What is the Antiderivative? In a derivative problem, a function f(x) is given and you find the derivative f (x) using

More information

{ independent variable some property or restriction about independent variable } where the vertical line is read such that.

{ independent variable some property or restriction about independent variable } where the vertical line is read such that. Page 1 of 5 Introduction to Review Materials One key to Algebra success is identifying the type of work necessary to answer a specific question. First you need to identify whether you are dealing with

More information

UNIT 4 NOTES: PROPERTIES & EXPRESSIONS

UNIT 4 NOTES: PROPERTIES & EXPRESSIONS UNIT 4 NOTES: PROPERTIES & EXPRESSIONS Vocabulary Mathematics: (from Greek mathema, knowledge, study, learning ) Is the study of quantity, structure, space, and change. Algebra: Is the branch of mathematics

More information

Algebra II Summer Packet. Summer Name:

Algebra II Summer Packet. Summer Name: Algebra II Summer Packet Summer 2017 Name: NAME ALGEBRA II & TRIGONOMETRY SUMMER REVIEW PACKET To maintain a high quality program, students entering Algebra II are expected to remember the basics of the

More information

Math 2 Variable Manipulation Part 2 Powers & Roots PROPERTIES OF EXPONENTS:

Math 2 Variable Manipulation Part 2 Powers & Roots PROPERTIES OF EXPONENTS: Math 2 Variable Manipulation Part 2 Powers & Roots PROPERTIES OF EXPONENTS: 1 EXPONENT REVIEW PROBLEMS: 2 1. 2x + x x + x + 5 =? 2. (x 2 + x) (x + 2) =?. The expression 8x (7x 6 x 5 ) is equivalent to?.

More information

Answers of the MATH97 Practice Test Form A

Answers of the MATH97 Practice Test Form A Answers of the MATH97 Practice Test Form A A1) Answer B Section 1.2: concepts of solution of the equations. Pick the pair which satisfies the equation 4x+y=10. x= 1 and y=6 A2) Answer A Section 1.3: select

More information

Solving Equations Quick Reference

Solving Equations Quick Reference Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number

More information

P.1: Algebraic Expressions, Mathematical Models, and Real Numbers

P.1: Algebraic Expressions, Mathematical Models, and Real Numbers Chapter P Prerequisites: Fundamental Concepts of Algebra Pre-calculus notes Date: P.1: Algebraic Expressions, Mathematical Models, and Real Numbers Algebraic expression: a combination of variables and

More information

Chapter 1 Indices & Standard Form

Chapter 1 Indices & Standard Form Chapter 1 Indices & Standard Form Section 1.1 Simplifying Only like (same letters go together; same powers and same letter go together) terms can be grouped together. Example: a 2 + 3ab + 4a 2 5ab + 10

More information

Section 1.3 Review of Complex Numbers

Section 1.3 Review of Complex Numbers 1 Section 1. Review of Complex Numbers Objective 1: Imaginary and Complex Numbers In Science and Engineering, such quantities like the 5 occur all the time. So, we need to develop a number system that

More information

Parenthesis and other grouping symbols. Exponential expressions. Multiplication & Division Addition & Subtraction.

Parenthesis and other grouping symbols. Exponential expressions. Multiplication & Division Addition & Subtraction. NAME SADDLE BROOK HIGH SCHOOL HONORS ALGEBRA II SUMMER PACKET To maintain a high quality program, students entering Honors Algebra II are expected to remember the basics of the mathematics taught in their

More information

Chapter 7 Rational Expressions, Equations, and Functions

Chapter 7 Rational Expressions, Equations, and Functions Chapter 7 Rational Expressions, Equations, and Functions Section 7.1: Simplifying, Multiplying, and Dividing Rational Expressions and Functions Section 7.2: Adding and Subtracting Rational Expressions

More information

Order of Operations P E M D A S. Notes: Expressions and Equations (6.EE.1 9) Exponents. Order of Operations x

Order of Operations P E M D A S. Notes: Expressions and Equations (6.EE.1 9) Exponents. Order of Operations x Parts: Exponents 5 Exponent Base Exponential Form Write the expression using a base and exponent. Expanded Form: Write out what the exponent means. x x x x x Standard Form: Solve the expression. 6 81 ***

More information

Chapter 5 Rational Expressions

Chapter 5 Rational Expressions Worksheet 4 (5.1 Chapter 5 Rational Expressions 5.1 Simplifying Rational Expressions Summary 1: Definitions and General Properties of Rational Numbers and Rational Expressions A rational number can be

More information

= (5) 1 3 = (4) 1 4 = (3) 1 5 = (2) 1 6

= (5) 1 3 = (4) 1 4 = (3) 1 5 = (2) 1 6 We have seen already when covering Lesson 3 that fractional exponents are simply an alternate way of expressing radicals. 6 = (6) 11 22 So a square root is equivalent to a power of 1, which is the reciprocal

More information

Equations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero

Equations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero Solving Other Types of Equations Rational Equations Examine each denominator to find values that would cause the denominator to equal zero Multiply each term by the LCD or If two terms cross-multiply Solve,

More information

Algebra 1 Unit 6 Notes

Algebra 1 Unit 6 Notes Algebra 1 Unit 6 Notes Name: Day Date Assignment (Due the next class meeting) Monday Tuesday Wednesday Thursday Friday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday

More information

6.1 Polynomial Functions

6.1 Polynomial Functions 6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and

More information

Part 2 - Beginning Algebra Summary

Part 2 - Beginning Algebra Summary Part - Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian

More information

MA094 Part 2 - Beginning Algebra Summary

MA094 Part 2 - Beginning Algebra Summary MA094 Part - Beginning Algebra Summary Page of 8/8/0 Big Picture Algebra is Solving Equations with Variables* Variable Variables Linear Equations x 0 MA090 Solution: Point 0 Linear Inequalities x < 0 page

More information

Mini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models

Mini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models Mini Lecture. Introduction to Algebra: Variables and Mathematical Models. Evaluate algebraic expressions.. Translate English phrases into algebraic expressions.. Determine whether a number is a solution

More information

P4 Polynomials and P5 Factoring Polynomials

P4 Polynomials and P5 Factoring Polynomials P4 Polynomials and P5 Factoring Polynomials Professor Tim Busken Graduate T.A. Dynamical Systems Program Department of Mathematics San Diego State University June 22, 2011 Professor Tim Busken (Graduate

More information

Sect Addition and Subtraction of Polynomials

Sect Addition and Subtraction of Polynomials Sect 5.5 - Addition and Subtraction of Polynomials Concept #1 Introduction to Polynomials Before we begin discussing polynomials, let s review some items from chapter 1 with the following example: Complete

More information

Section 2.4: Add and Subtract Rational Expressions

Section 2.4: Add and Subtract Rational Expressions CHAPTER Section.: Add and Subtract Rational Expressions Section.: Add and Subtract Rational Expressions Objective: Add and subtract rational expressions with like and different denominators. You will recall

More information

Unit 13: Polynomials and Exponents

Unit 13: Polynomials and Exponents Section 13.1: Polynomials Section 13.2: Operations on Polynomials Section 13.3: Properties of Exponents Section 13.4: Multiplication of Polynomials Section 13.5: Applications from Geometry Section 13.6:

More information

MATH 190 KHAN ACADEMY VIDEOS

MATH 190 KHAN ACADEMY VIDEOS MATH 10 KHAN ACADEMY VIDEOS MATTHEW AUTH 11 Order of operations 1 The Real Numbers (11) Example 11 Worked example: Order of operations (PEMDAS) 7 2 + (7 + 3 (5 2)) 4 2 12 Rational + Irrational Example

More information

Use properties of exponents. Use the properties of rational exponents to simplify the expression. 12 d.

Use properties of exponents. Use the properties of rational exponents to simplify the expression. 12 d. EXAMPLE 1 Use properties of exponents Use the properties of rational exponents to simplify the expression. a. 7 1/ 7 1/2 7 (1/ + 1/2) 7 / b. (6 1/2 1/ ) 2 (6 1/2 ) 2 ( 1/ ) 2 6( 1/2 2 ) ( 1/ 2 ) 6 1 2/

More information

Accuplacer Review Workshop. Elementary Algebra Part II. Week Three. Includes internet links to instructional videos for additional resources:

Accuplacer Review Workshop. Elementary Algebra Part II. Week Three. Includes internet links to instructional videos for additional resources: Accuplacer Review Workshop Elementary Algebra Part II Week Three Includes internet links to instructional videos for additional resources: http://www.mathispower4u.com (Arithmetic Video Library) http://www.purplemath.com

More information

SOLUTIONS FOR PROBLEMS 1-30

SOLUTIONS FOR PROBLEMS 1-30 . Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).

More information

The trick is to multiply the numerator and denominator of the big fraction by the least common denominator of every little fraction.

The trick is to multiply the numerator and denominator of the big fraction by the least common denominator of every little fraction. Complex Fractions A complex fraction is an expression that features fractions within fractions. To simplify complex fractions, we only need to master one very simple method. Simplify 7 6 +3 8 4 3 4 The

More information

UNIT 4: RATIONAL AND RADICAL EXPRESSIONS. 4.1 Product Rule. Objective. Vocabulary. o Scientific Notation. o Base

UNIT 4: RATIONAL AND RADICAL EXPRESSIONS. 4.1 Product Rule. Objective. Vocabulary. o Scientific Notation. o Base UNIT 4: RATIONAL AND RADICAL EXPRESSIONS M1 5.8, M2 10.1-4, M3 5.4-5, 6.5,8 4.1 Product Rule Objective I will be able to multiply powers when they have the same base, including simplifying algebraic expressions

More information

Algebra 2 Honors: Final Exam Review

Algebra 2 Honors: Final Exam Review Name: Class: Date: Algebra 2 Honors: Final Exam Review Directions: You may write on this review packet. Remember that this packet is similar to the questions that you will have on your final exam. Attempt

More information

CD't1D Find 4 + (-6). ~ Find -2 + (-3).

CD't1D Find 4 + (-6). ~ Find -2 + (-3). Add Integers To add integers with the same sign, add their absolute values. The sum is: positive if both integers are positive. negative if both integers are negative. To add integers with different signs,

More information

UNIT 5 EXPONENTS NAME: PERIOD:

UNIT 5 EXPONENTS NAME: PERIOD: NAME: PERIOD: UNIT 5 EXPONENTS Disclaimer: This packet is your notes for all of unit 5. It is expected you will take good notes and work the examples in class with your teacher in pencil. It is expected

More information

Study Guide for Math 095

Study Guide for Math 095 Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.

More information

A field trips costs $800 for the charter bus plus $10 per student for x students. The cost per student is represented by: 10x x

A field trips costs $800 for the charter bus plus $10 per student for x students. The cost per student is represented by: 10x x LEARNING STRATEGIES: Activate Prior Knowledge, Shared Reading, Think/Pair/Share, Note Taking, Group Presentation, Interactive Word Wall A field trips costs $800 for the charter bus plus $10 per student

More information

Unit 9 Study Sheet Rational Expressions and Types of Equations

Unit 9 Study Sheet Rational Expressions and Types of Equations Algebraic Fractions: Unit 9 Study Sheet Rational Expressions and Types of Equations Simplifying Algebraic Fractions: To simplify an algebraic fraction means to reduce it to lowest terms. This is done by

More information

Sec 2.1 The Real Number Line. Opposites: Two numbers that are the same distance from the origin (zero), but on opposite sides of the origin.

Sec 2.1 The Real Number Line. Opposites: Two numbers that are the same distance from the origin (zero), but on opposite sides of the origin. Algebra 1 Chapter 2 Note Packet Name Sec 2.1 The Real Number Line Real Numbers- All the numbers on the number line, not just whole number integers (decimals, fractions and mixed numbers, square roots,

More information

Algebra 2 and Trigonometry

Algebra 2 and Trigonometry Algebra 2 and Trigonometry Chapter 7: Exponential Functions Name: Teacher: Pd: 1 Table of Contents Day 1: Chapter 7-1/7-2: Laws of Exponents SWBAT: Simplify positive, negative, and zero exponents. Pgs.

More information

Sample Math 21 Exam Questions No Calculators Allowed

Sample Math 21 Exam Questions No Calculators Allowed No Calculators Allowed 1. Remove all grouping symbols and combine like terms: (a) (3a 4) + ( 5a + 6a) (a 7a + 11) (b) {5x 3y + [x (5x 7y)] + 4y} a [b + (a 4b)] + [5a 3b]. Perform the indicated multiplication

More information

Exponents, Polynomials, and Polynomial Functions. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 1

Exponents, Polynomials, and Polynomial Functions. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 1 5 Exponents, Polynomials, and Polynomial Functions Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 1 5.1 Integer Exponents R.1 Fractions and Scientific Notation Objectives 1. Use the product

More information

CD't1D Find 4 + (-6). ~ Find -2 + (-3).

CD't1D Find 4 + (-6). ~ Find -2 + (-3). Add Integers To add integers with the same sign, add their absolute values. The sum is: positive if both integers are positive. negative if both integers are negative. To add integers with different signs,

More information

Beginning Algebra MAT0024C. Professor Sikora. Professor M. J. Sikora ~ Valencia Community College

Beginning Algebra MAT0024C. Professor Sikora. Professor M. J. Sikora ~ Valencia Community College Beginning Algebra Professor Sikora MAT002C POLYNOMIALS 6.1 Positive Integer Exponents x n = x x x x x [n of these x factors] base exponent Numerical: Ex: - = where as Ex: (-) = Ex: - = and Ex: (-) = Rule:

More information

Math Lesson 2-2 Properties of Exponents

Math Lesson 2-2 Properties of Exponents Math-00 Lesson - Properties of Eponents Properties of Eponents What is a power? Power: An epression formed b repeated multiplication of the base. Coefficient Base Eponent The eponent applies to the number

More information

Polynomial Functions

Polynomial Functions Polynomial Functions Polynomials A Polynomial in one variable, x, is an expression of the form a n x 0 a 1 x n 1... a n 2 x 2 a n 1 x a n The coefficients represent complex numbers (real or imaginary),

More information

= ( 17) = (-4) + (-6) = (-3) + (- 14) + 20

= ( 17) = (-4) + (-6) = (-3) + (- 14) + 20 Integer Operations Adding Integers If the signs are the same, add the numbers and keep the sign. If the signs are different, find the difference and use the sign of the number with the greatest absolute

More information

Adding and Subtracting Rational Expressions. Add and subtract rational expressions with the same denominator.

Adding and Subtracting Rational Expressions. Add and subtract rational expressions with the same denominator. Chapter 7 Section 7. Objectives Adding and Subtracting Rational Expressions 1 3 Add and subtract rational expressions with the same denominator. Find a least common denominator. Add and subtract rational

More information

Class 8: Numbers Exercise 3B

Class 8: Numbers Exercise 3B Class : Numbers Exercise B 1. Compare the following pairs of rational numbers: 1 1 i First take the LCM of. LCM = 96 Therefore: 1 = 96 Hence we see that < 6 96 96 1 1 1 1 = 6 96 1 or we can say that

More information

MathB65 Ch 4 IV, V, VI.notebook. October 31, 2017

MathB65 Ch 4 IV, V, VI.notebook. October 31, 2017 Part 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest

More information

Sect Addition, Subtraction, Multiplication, and Division Properties of Equality

Sect Addition, Subtraction, Multiplication, and Division Properties of Equality Sect.1 - Addition, Subtraction, Multiplication, and Division Properties of Equality Concept #1 Definition of a Linear Equation in One Variable An equation is a statement that two quantities are equal.

More information

The most factored form is usually accomplished by common factoring the expression. But, any type of factoring may come into play.

The most factored form is usually accomplished by common factoring the expression. But, any type of factoring may come into play. MOST FACTORED FORM The most factored form is the most factored version of a rational expression. Being able to find the most factored form is an essential skill when simplifying the derivatives found using

More information

Algebra 1 Summer Assignment 2018

Algebra 1 Summer Assignment 2018 Algebra 1 Summer Assignment 2018 The following packet contains topics and definitions that you will be required to know in order to succeed in Algebra 1 this coming school year. You are advised to be familiar

More information

JUST THE MATHS UNIT NUMBER 1.3. ALGEBRA 3 (Indices and radicals (or surds)) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.3. ALGEBRA 3 (Indices and radicals (or surds)) A.J.Hobson JUST THE MATHS UNIT NUMBER 1 ALGEBRA (Indices and radicals (or surds)) by AJHobson 11 Indices 12 Radicals (or Surds) 1 Exercises 14 Answers to exercises UNIT 1 - ALGEBRA - INDICES AND RADICALS (or Surds)

More information

Numbers and Operations Review

Numbers and Operations Review C H A P T E R 5 Numbers and Operations Review This chapter reviews key concepts of numbers and operations that you need to know for the SAT. Throughout the chapter are sample questions in the style of

More information

CHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section Multiplying and Dividing Rational Expressions

CHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section Multiplying and Dividing Rational Expressions Name Objectives: Period CHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section 8.3 - Multiplying and Dividing Rational Expressions Multiply and divide rational expressions. Simplify rational expressions,

More information

SYMBOL NAME DESCRIPTION EXAMPLES. called positive integers) negatives, and 0. represented as a b, where

SYMBOL NAME DESCRIPTION EXAMPLES. called positive integers) negatives, and 0. represented as a b, where EXERCISE A-1 Things to remember: 1. THE SET OF REAL NUMBERS SYMBOL NAME DESCRIPTION EXAMPLES N Natural numbers Counting numbers (also 1, 2, 3,... called positive integers) Z Integers Natural numbers, their

More information

Linear Equations & Inequalities Definitions

Linear Equations & Inequalities Definitions Linear Equations & Inequalities Definitions Constants - a term that is only a number Example: 3; -6; -10.5 Coefficients - the number in front of a term Example: -3x 2, -3 is the coefficient Variable -

More information

Review for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.

Review for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition. LESSON 6- Review for Mastery Integer Exponents Remember that means 8. The base is, the exponent is positive. Exponents can also be 0 or negative. Zero Exponents Negative Exponents Negative Exponents in

More information

We will work with two important rules for radicals. We will write them for square roots but they work for any root (cube root, fourth root, etc.).

We will work with two important rules for radicals. We will write them for square roots but they work for any root (cube root, fourth root, etc.). College algebra We will review simplifying radicals, exponents and their rules, multiplying polynomials, factoring polynomials, greatest common denominators, and solving rational equations. Pre-requisite

More information

2015 SUMMER MATH PACKET

2015 SUMMER MATH PACKET Name: Date: 05 SUMMER MATH PACKET College Algebra Trig. - I understand that the purpose of the summer packet is for my child to review the topics they have already mastered in previous math classes and

More information

Ch. 12 Rational Functions

Ch. 12 Rational Functions Ch. 12 Rational Functions 12.1 Finding the Domains of Rational F(n) & Reducing Rational Expressions Outline Review Rational Numbers { a / b a and b are integers, b 0} Multiplying a rational number by a

More information

( ) c. m = 0, 1 2, 3 4

( ) c. m = 0, 1 2, 3 4 G Linear Functions Probably the most important concept from precalculus that is required for differential calculus is that of linear functions The formulas you need to know backwards and forwards are:

More information

Tuesday, 3/28 : Ch. 9.8 Cubic Functions ~ Ch. 9 Packet p.67 #(1-6) Thursday, 3/30 : Ch. 9.8 Rational Expressions ~ Ch. 9 Packet p.

Tuesday, 3/28 : Ch. 9.8 Cubic Functions ~ Ch. 9 Packet p.67 #(1-6) Thursday, 3/30 : Ch. 9.8 Rational Expressions ~ Ch. 9 Packet p. Ch. 9.8 Cubic Functions & Ch. 9.8 Rational Expressions Learning Intentions: Explore general patterns & characteristics of cubic functions. Learn formulas that model the areas of squares & the volumes of

More information

LESSON 9.1 ROOTS AND RADICALS

LESSON 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 information

Day 3: Section P-6 Rational Expressions; Section P-7 Equations. Rational Expressions

Day 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 information

P.1 Prerequisite skills Basic Algebra Skills

P.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 information

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2

Summer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2 Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level

More information

Rising 8th Grade Math. Algebra 1 Summer Review Packet

Rising 8th Grade Math. Algebra 1 Summer Review Packet Rising 8th Grade Math Algebra 1 Summer Review Packet 1. Clear parentheses using the distributive property. 2. Combine like terms within each side of the equal sign. Solving Multi-Step Equations 3. Add/subtract

More information

Math 005A Prerequisite Material Answer Key

Math 005A Prerequisite Material Answer Key Math 005A Prerequisite Material Answer Key 1. a) P = 4s (definition of perimeter and square) b) P = l + w (definition of perimeter and rectangle) c) P = a + b + c (definition of perimeter and triangle)

More information

CONTENTS COLLEGE ALGEBRA: DR.YOU

CONTENTS COLLEGE ALGEBRA: DR.YOU 1 CONTENTS CONTENTS Textbook UNIT 1 LECTURE 1-1 REVIEW A. p. LECTURE 1- RADICALS A.10 p.9 LECTURE 1- COMPLEX NUMBERS A.7 p.17 LECTURE 1-4 BASIC FACTORS A. p.4 LECTURE 1-5. SOLVING THE EQUATIONS A.6 p.

More information

Math-2. Lesson:1-2 Properties of Exponents

Math-2. Lesson:1-2 Properties of Exponents Math- Lesson:- Properties of Eponents Properties of Eponents What is a power? Power: An epression formed b repeated multiplication of the same factor. Coefficient Base Eponent The eponent applies to the

More information

LESSON 6.1 EXPONENTS LESSON 6.1 EXPONENTS 253

LESSON 6.1 EXPONENTS LESSON 6.1 EXPONENTS 253 LESSON 6.1 EXPONENTS LESSON 6.1 EXPONENTS 5 OVERVIEW Here's what you'll learn in this lesson: Properties of Exponents Definition of exponent, power, and base b. Multiplication Property c. Division Property

More information

Final Exam Review for DMAT 0310

Final Exam Review for DMAT 0310 Final Exam Review for DMAT 010 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Factor the polynomial completely. What is one of the factors? 1) x

More information

8.1 Multiplication Properties of Exponents Objectives 1. Use properties of exponents to multiply exponential expressions.

8.1 Multiplication Properties of Exponents Objectives 1. Use properties of exponents to multiply exponential expressions. 8.1 Multiplication Properties of Exponents Objectives 1. Use properties of exponents to multiply exponential expressions. 2. Use powers to model real life problems. Multiplication Properties of Exponents

More information

8.1 Apply Exponent Properties Involving Products. Learning Outcome To use properties of exponents involving products

8.1 Apply Exponent Properties Involving Products. Learning Outcome To use properties of exponents involving products 8.1 Apply Exponent Properties Involving Products Learning Outcome To use properties of exponents involving products Product of Powers Property Let a be a real number, and let m and n be positive integers.

More information

Chapter 1: Fundamentals of Algebra Lecture notes Math 1010

Chapter 1: Fundamentals of Algebra Lecture notes Math 1010 Section 1.1: The Real Number System Definition of set and subset A set is a collection of objects and its objects are called members. If all the members of a set A are also members of a set B, then A is

More information

Order of Operations. Real numbers

Order of Operations. Real numbers Order of Operations When simplifying algebraic expressions we use the following order: 1. Perform operations within a parenthesis. 2. Evaluate exponents. 3. Multiply and divide from left to right. 4. Add

More information

Section 2.3 Solving Linear Equations

Section 2.3 Solving Linear Equations Variable: Defined as A symbol (or letter) that is used to represent an unknown numbers Examples: a, b, c, x, y, z, s, t, m, n, Constant: Defined as A single number Examples: 1, 2, 3, 6, 1, π, e, π, 1.6,

More information

Sect Introduction to Rational Expressions

Sect Introduction to Rational Expressions 127 Sect 7.1 - Introduction to Rational Expressions Concept #1 Definition of a Rational Expression. Recall that a rational number is any number that can be written as the ratio of two integers where the

More information

1.2. Indices. Introduction. Prerequisites. Learning Outcomes

1.2. Indices. Introduction. Prerequisites. Learning Outcomes Indices.2 Introduction Indices, or powers, provide a convenient notation when we need to multiply a number by itself several times. In this section we explain how indices are written, and state the rules

More information