# Negative Exponents and Scientific Notation

Size: px
Start display at page:

Transcription

1 3.2 Negative Exponents and Scientific Notation 3.2 OBJECTIVES. Evaluate expressions involving zero or a negative exponent 2. Simplify expressions involving zero or a negative exponent 3. Write a decimal number in scientific notation 4. Solve an application of scientific notation NOTE By Property 2, a m a n am n when m n. Here m and n are both 5 so m n. In Section 3., we discussed exponents. We now want to extend our exponent notation to include 0 and negative integers as exponents. First, what do we do with x 0? It will help to look at a problem that gives us x 0 as a result. What if the numerator and denominator of a fraction have the same base raised to the same power and we extend our division rule? For example, a 5 () a 5 a5 5 a 0 But from our experience with fractions we know that a 5 (2) a 5 By comparing equations () and (2), it seems reasonable to make the following definition: 0 NOTE As was the case with, will be discussed in a later course. Definitions: Zero Power For any number a, a 0, a 0 In words, any expression, except 0, raised to the 0 power is. Example illustrates the use of this definition. Example Raising Expressions to the Zero Power CAUTION In part (d ) the 0 exponent applies only to the x and not to the factor 6, because the base is x. Evaluate. Assume all variables are nonzero. (a) 5 0 (b) 27 0 (c) (x 2 y) 0 if x 0 and y 0 (d) 6x if x 0 CHECK YOURSELF Evaluate. Assume all variables are nonzero. (a) 7 0 (b) ( 8) 0 (c) (xy 3 ) 0 (d) 3x 0 26

2 262 CHAPTER 3 POLYNOMIALS The second property of exponents allows us to define a negative exponent. Suppose that the exponent in the denominator is greater than the exponent in the numerator. Consider x 2 x 5 the expression. Our previous work with fractions tells us that NOTE Divide the numerator and denominator by the two common factors of x. REMEMBER: am a n am n x 2 x 5 x x x x x x x x 3 However, if we extend the second property to let n be greater than m, we have x 2 x 5 x2 5 x 3 Now, by comparing equations () and (2), it seems reasonable to define x 3 as. x In general, we have this result: 3 () (2) Definitions: Negative Powers NOTE John Wallis (66 703), an English mathematician, was the first to fully discuss the meaning of 0 and negative exponents. For any number a, a 0, and any positive integer n, a n a n Example 2 Rewriting Expressions That Contain Negative Exponents Rewrite each expression, using only positive exponents. Negative exponent in numerator (a) x 4 x 4 Positive exponent in denominator (b) m 7 m 7 (c) or 9 (d) or 000 CAUTION (e) (f) 2x 3 a 5 a 9 2 x 3 2 x 3 The 3 exponent applies only to x, because x is the base. a 5 9 a 4 a 4 (g) 4x 5 4 x 5 4 x 5

3 NEGATIVE EXPONENTS AND SCIENTIFIC NOTATION SECTION CHECK YOURSELF 2 Write, using only positive exponents. (a) a 0 (b) 4 3 (c) 3x 2 (d) x5 x 8 We will now allow negative integers as exponents in our first property for exponents. Consider Example 3. Example 3 Simplifying Expressions Containing Exponents NOTE a m a n a m n for any integers m and n. So add the exponents. Simplify (write an equivalent expression that uses only positive exponents). (a) x 5 x 2 x 5 ( 2) x 3 Note: An alternative approach would be NOTE By definition x 2 x 2 x 5 x 2 x 5 x 2 x5 x 2 x3 (b) a 7 a 5 a 7 ( 5) a 2 (c) y 5 y 9 y 5 ( 9) y 4 y 4 CHECK YOURSELF 3 Simplify (write an equivalent expression that uses only positive exponents). (a) x 7 x 2 (b) b 3 b 8 Example 4 shows that all the properties of exponents introduced in the last section can be extended to expressions with negative exponents. Example 4 Simplifying Expressions Containing Exponents Simplify each expression. (a) (b) m 3 m 4 a 2 b 6 a 5 b 4 m 3 4 m 7 m 7 a 2 5 b 6 ( 4) a 7 b 0 b0 a 7 Property 2 Apply Property 2 to each variable.

4 264 CHAPTER 3 POLYNOMIALS NOTE This could also be done by using Property 4 first, so (c) (2x 4 ) 3 (2x 4 ) 3 Definition of the negative exponent (2x 4 ) (x 4 ) x x 2 8x (x 4 ) 3 8x 2 Property 4 Property 3 (d) (y 2 ) 4 (y 3 ) 2 y 8 y 6 y 8 ( 6) Property 3 Property 2 y 2 y 2 CHECK YOURSELF 4 Simplify each expression. x 5 m 3 n 5 (a) (b) (c) (3a 3 ) 4 (d) (r3 ) 2 x 3 m 2 n 3 (r 4 ) 2 Let us now take a look at an important use of exponents, scientific notation. We begin the discussion with a calculator exercise. On most calculators, if you multiply 2.3 times 000, the display will read 2300 Multiply by 000 a second time. Now you will see Multiplying by 000 a third time will result in the display NOTE This must equal 2,300,000, or 2.3 E09 And multiplying by 000 again yields NOTE Consider the following table: , , or 2.3 E2 Can you see what is happening? This is the way calculators display very large numbers. The number on the left is always between and 0, and the number on the right indicates the number of places the decimal point must be moved to the right to put the answer in standard (or decimal) form. This notation is used frequently in science. It is not uncommon in scientific applications of algebra to find yourself working with very large or very small numbers. Even in the time of Archimedes ( B.C.E.), the study of such numbers was not unusual. Archimedes estimated that the universe was 23,000,000,000,000,000 m in diameter, which is the approximate distance light travels in 2 years. By comparison, Polaris (the North Star) is 2 actually 680 light-years from the earth. Example 6 will discuss the idea of light-years.

5 NEGATIVE EXPONENTS AND SCIENTIFIC NOTATION SECTION In scientific notation, Archimedes s estimate for the diameter of the universe would be m In general, we can define scientific notation as follows. Definitions: Scientific Notation Any number written in the form a 0 n in which a 0 and n is an integer, is written in scientific notation. Example 5 Using Scientific Notation Write each of the following numbers in scientific notation. NOTE Notice the pattern for writing a number in scientific notation. (a) 20, places (b) 88,000, The power is 5. 7 places The power is 7. NOTE The exponent on 0 shows the number of places we must move the decimal point. A positive exponent tells us to move right, and a negative exponent indicates to move left. (c) 520,000, places (d) 4,000,000, places (e) places If the decimal point is to be moved to the left, the exponent will be negative. NOTE To convert back to standard or decimal form, the process is simply reversed. (f) places CHECK YOURSELF 5 Write in scientific notation. (a) 22,000,000,000,000,000 (b) (c) 5,600,000 (d)

6 266 CHAPTER 3 POLYNOMIALS Example 6 An Application of Scientific Notation (a) Light travels at a speed of meters per second (m/s). There are approximately s in a year. How far does light travel in a year? We multiply the distance traveled in s by the number of seconds in a year. This yields NOTE Notice that ( )( ) ( )( ) Multiply the coefficients, and add the exponents. For our purposes we round the distance light travels in year to 0 6 m. This unit is called a light-year, and it is used to measure astronomical distances. (b) The distance from earth to the star Spica (in Virgo) is m. How many lightyears is Spica from earth? Spica m Earth NOTE We divide the distance (in meters) by the number of meters in light-year light-years CHECK YOURSELF 6 The farthest object that can be seen with the unaided eye is the Andromeda galaxy. This galaxy is m from earth. What is this distance in light-years? CHECK YOURSELF ANSWERS 3. (a) ; (b) ; (c) ; (d) 3 2. (a) ; (b) ; (c) ; (d) 4 3 or a 0 64 x 2 x 3 m 5 3. (a) x 5 ; (b) 4. (a) x 8 ; (b) ; (c) ; (d) r 2 b 5 n 8 8a 2 5. (a) ; (b) ; (c) ; (d) ,300,000 light-years

7 Name 3.2 Exercises Section Date Evaluate (assume the variables are nonzero) ( 7) 0 3. ( 29) (x 3 y 2 ) m 0 7. x 0 8. (2a 3 b 7 ) 0 9. ( 3p 6 q 8 ) x 0 Write each of the following expressions using positive exponents; simplify when possible. ANSWERS b 8 2. p x a 2 2. (5x) 22. (3a) x x ( 2x) (3x) Use Properties and 2 to simplify each of the following expressions. Write your answers with positive exponents only. 27. a 5 a m 5 m x 8 x a 2 a 8 3. b 7 b 32. y 5 y x 0 x r 3 r a 8 a

8 ANSWERS m 9 m r r 5 x 7 x 9 x 3 x 5 a 3 a 0 x 4 x p 6 p Simplify each of the following expressions. Write your answers with positive exponents only. m 5 n 3 p 3 q (2a 3 ) 4 m 4 n 5 p 4 q (3x 2 ) (x 2 y 3 ) (a 5 b 3 ) (r 2 ) 3 (y 3 ) r 4 (m 4 ) 3 (a 3 ) 2 (a 4 ) (m 2 ) 4 (a 3 ) 3 y 6 (x 3 ) 3 (x 4 ) 2 (x 2 ) 3 (x 2 ) (x 2 ) In exercises 55 to 58, express each number in scientific notation The distance from the earth to the sun: 93,000,000 mi The diameter of a grain of sand: m. 57. The diameter of the sun: 30,000,000,000 cm. 58. The number of molecules in 22.4 L of a gas: 602,000,000,000,000,000,000,000 (Avogadro s number). 59. The mass of the sun is approximately kg. If this were written in standard or decimal form, how many 0s would follow the digit 8? 268

9 ANSWERS 60. Archimedes estimated the universe to be millimeters (mm) in diameter. If this number were written in standard or decimal form, how many 0s would follow the digit 3? In exercises 6 to 64, write each expression in standard notation In exercises 65 to 68, write each of the following in scientific notation In exercises 69 to 72, compute the expressions using scientific notation, and write your answer in that form. 69. (4 0 3 )(2 0 5 ) 70. ( )(4 0 2 ) In exercises 73 to 78, perform the indicated calculations. Write your result in scientific notation (2 0 5 )(4 0 4 ) 74. ( )(3 0 5 ) ( )(6 0 5 ) (. 0 8 )(3 0 6 ) In 975 the population of Earth was approximately 4 billion and doubling every 35 years. The formula for the population P in year Y for this doubling rate is P (in billions) 4 2 (Y 975) What was the approximate population of Earth in 960? 80. What will Earth s population be in 2025? The United States population in 990 was approximately 250 million, and the average growth rate for the past 30 years gives a doubling time of 66 years. The above formula for the United States then becomes P (in millions) (Y 990) What was the approximate population of the United States in 960? (6 0 2 )( ) ( )(3 0 2 ) 82. What will be the population of the United States in 2025 if this growth rate continues?

10 ANSWERS a. b. c. d. e. f. g. h. 83. Megrez, the nearest of the Big Dipper stars, is m from Earth. Approximately how long does it take light, traveling at 0 6 m/year, to travel from Megrez to Earth? 84. Alkaid, the most distant star in the Big Dipper, is m from Earth. Approximately how long does it take light to travel from Alkaid to Earth? 85. The number of liters (L) of water on Earth is 5,500 followed by 9 zeros. Write this number in scientific notation. Then use the number of liters of water on Earth to find out how much water is available for each person on Earth. The population of Earth is 6 billion. 86. If there are people on Earth and there is enough freshwater to provide each person with L, how much freshwater is on Earth? 87. The United States uses an average of L of water per person each year. The United States has people. How many liters of water does the United States use each year? Getting Ready for Section 3.3 [Section.6] Combine like terms where possible. (a) 8m 7m (b) 9x 5x (c) 9m 2 8m (d) 8x 2 7x 2 (e) 5c 3 5c 3 (f) 9s 3 8s 3 (g) 8c 2 6c 2c 2 (h) 8r 3 7r 2 5r 3 Answers b ,000 x 5x x a x x a x 5 b 4 x m 9 6 x 4 4. x r 8 n 8 a 2 y 6 r 2 x 53. a mi cm billion million years L; L L a. 5m b. 4x c. 9m 2 8m d. x 2 e. 20c 3 f. 7s 3 g. 0c 2 6c h. 3r 3 7r 2 270

### Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m

0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have

### Negative Integer Exponents

7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions

### Session 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers:

Session 9 Scientific Notation and Laws of Exponents If you have ever taken a Chemistry class, you may have encountered the following numbers: There are approximately 60,4,79,00,000,000,000,000 molecules

### Exponents, 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) =

### Rational 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

### Multiplying Fractions

. Multiplying Fractions. OBJECTIVES 1. Multiply two fractions. Multiply two mixed numbers. Simplify before multiplying fractions 4. Estimate products by rounding Multiplication is the easiest of the four

### A positive exponent means repeated multiplication. A negative exponent means the opposite of repeated multiplication, which is repeated

Eponents Dealing with positive and negative eponents and simplifying epressions dealing with them is simply a matter of remembering what the definition of an eponent is. division. A positive eponent means

### 2.2 Scientific Notation: Writing Large and Small Numbers

2.2 Scientific Notation: Writing Large and Small Numbers A number written in scientific notation has two parts. A decimal part: a number that is between 1 and 10. An exponential part: 10 raised to an exponent,

### A.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

### How 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

### Exponents. Learning Objectives 4-1

Eponents -1 to - Learning Objectives -1 The product rule for eponents The quotient rule for eponents The power rule for eponents Power rules for products and quotient We can simplify by combining the like

### Negative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2

4 (4-) Chapter 4 Polynomials and Eponents P( r) 0 ( r) dollars. Which law of eponents can be used to simplify the last epression? Simplify it. P( r) 7. CD rollover. Ronnie invested P dollars in a -year

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### DIMENSIONAL ANALYSIS #2

DIMENSIONAL ANALYSIS #2 Area is measured in square units, such as square feet or square centimeters. These units can be abbreviated as ft 2 (square feet) and cm 2 (square centimeters). For example, we

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### To 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

8. Radicals - Rational Exponents Objective: Convert between radical notation and exponential notation and simplify expressions with rational exponents using the properties of exponents. When we simplify

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### Review of Scientific Notation and Significant Figures

II-1 Scientific Notation Review of Scientific Notation and Significant Figures Frequently numbers that occur in physics and other sciences are either very large or very small. For example, the speed of

### Simplifying Algebraic Fractions

5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions

### Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

### Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

### The gas can has a capacity of 4.17 gallons and weighs 3.4 pounds.

hundred million\$ ten------ million\$ million\$ 00,000,000 0,000,000,000,000 00,000 0,000,000 00 0 0 0 0 0 0 0 0 0 Session 26 Decimal Fractions Explain the meaning of the values stated in the following sentence.

### SOLVING 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

### Scientific Notation. Section 7-1 Part 2

Scientific Notation Section 7-1 Part 2 Goals Goal To write numbers in scientific notation and standard form. To compare and order numbers using scientific notation. Vocabulary Scientific Notation Powers

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

### This 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

### Scales of the Universe

29:50 Astronomy Lab Stars, Galaxies, and the Universe Name Partner(s) Date Grade Category Max Points Points Received On Time 5 Printed Copy 5 Lab Work 90 Total 100 Scales of the Universe 1. Introduction

### Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

### PREPARATION FOR MATH TESTING at CityLab Academy

PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST

### Exponents. Exponents tell us how many times to multiply a base number by itself.

Exponents Exponents tell us how many times to multiply a base number by itself. Exponential form: 5 4 exponent base number Expanded form: 5 5 5 5 25 5 5 125 5 625 To use a calculator: put in the base number,

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### Exponential Notation and the Order of Operations

1.7 Exponential Notation and the Order of Operations 1.7 OBJECTIVES 1. Use exponent notation 2. Evaluate expressions containing powers of whole numbers 3. Know the order of operations 4. Evaluate expressions

### Multiplying and Dividing Signed Numbers. Finding the Product of Two Signed Numbers. (a) (3)( 4) ( 4) ( 4) ( 4) 12 (b) (4)( 5) ( 5) ( 5) ( 5) ( 5) 20

SECTION.4 Multiplying and Dividing Signed Numbers.4 OBJECTIVES 1. Multiply signed numbers 2. Use the commutative property of multiplication 3. Use the associative property of multiplication 4. Divide signed

### Quick 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

### Rules of Exponents. Math at Work: Motorcycle Customization OUTLINE CHAPTER

Rules of Exponents CHAPTER 5 Math at Work: Motorcycle Customization OUTLINE Study Strategies: Taking Math Tests 5. Basic Rules of Exponents Part A: The Product Rule and Power Rules Part B: Combining the

### Chapter 1 Lecture Notes: Science and Measurements

Educational Goals Chapter 1 Lecture Notes: Science and Measurements 1. Explain, compare, and contrast the terms scientific method, hypothesis, and experiment. 2. Compare and contrast scientific theory

### Free 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

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### Chapter 2 Measurement and Problem Solving

Introductory Chemistry, 3 rd Edition Nivaldo Tro Measurement and Problem Solving Graph of global Temperature rise in 20 th Century. Cover page Opposite page 11. Roy Kennedy Massachusetts Bay Community

### Figure 1. A typical Laboratory Thermometer graduated in C.

SIGNIFICANT FIGURES, EXPONENTS, AND SCIENTIFIC NOTATION 2004, 1990 by David A. Katz. All rights reserved. Permission for classroom use as long as the original copyright is included. 1. SIGNIFICANT FIGURES

### Solutions of Linear Equations in One Variable

2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

### Math 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

### Chapter 3 Review Math 1030

Section A.1: Three Ways of Using Percentages Using percentages We can use percentages in three different ways: To express a fraction of something. For example, A total of 10, 000 newspaper employees, 2.6%

### MBA Jump Start Program

MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right

### CHAPTER 4 DIMENSIONAL ANALYSIS

CHAPTER 4 DIMENSIONAL ANALYSIS 1. DIMENSIONAL ANALYSIS Dimensional analysis, which is also known as the factor label method or unit conversion method, is an extremely important tool in the field of chemistry.

### Student Exploration: Unit Conversions

Name: Date: Student Exploration: Unit Conversions Vocabulary: base unit, cancel, conversion factor, dimensional analysis, metric system, prefix, scientific notation Prior Knowledge Questions (Do these

### COWLEY 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

### Numerator Denominator

Fractions A fraction is any part of a group, number or whole. Fractions are always written as Numerator Denominator A unitary fraction is one where the numerator is always 1 e.g 1 1 1 1 1...etc... 2 3

### LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,

LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are

### Definition 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

### Overview for Families

unit: Ratios and Rates Mathematical strand: Number The following pages will help you to understand the mathematics that your child is currently studying as well as the type of problems (s)he will solve

### Chapter 1: Order of Operations, Fractions & Percents

HOSP 1107 (Business Math) Learning Centre Chapter 1: Order of Operations, Fractions & Percents ORDER OF OPERATIONS When finding the value of an expression, the operations must be carried out in a certain

### 26 Integers: Multiplication, Division, and Order

26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue

### 1.3 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,

### Solving Rational Equations and Inequalities

8-5 Solving Rational Equations and Inequalities TEKS 2A.10.D Rational functions: determine the solutions of rational equations using graphs, tables, and algebraic methods. Objective Solve rational equations

### Solving Equations With Fractional Coefficients

Solving Equations With Fractional Coefficients Some equations include a variable with a fractional coefficient. Solve this kind of equation by multiplying both sides of the equation by the reciprocal of

### Week 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

### SIMPLIFYING ALGEBRAIC FRACTIONS

Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is

### SIMPLIFYING 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

### Welcome to Physics 40!

Welcome to Physics 40! Physics for Scientists and Engineers Lab 1: Introduction to Measurement SI Quantities & Units In mechanics, three basic quantities are used Length, Mass, Time Will also use derived

### Multiplication 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

### 2.3 Solving Equations Containing Fractions and Decimals

2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions

### MATH 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

### Florida 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

### When factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF.

Factoring: reversing the distributive property. The distributive property allows us to do the following: When factoring, we look for greatest common factor of each term and reverse the distributive property

### Florida 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

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

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

### Chapter 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

### Zeros 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

### 1.3 Polynomials and Factoring

1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

### Exponents and Polynomials

Exponents and Polynomials 7A Exponents 7-1 Integer Exponents 7-2 Powers of 10 and Scientific Notation Lab Explore Properties of Exponents 7-3 Multiplication Properties of Exponents 7-4 Division Properties

### MATH-0910 Review Concepts (Haugen)

Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,

### Number Sense and Operations

Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents

### Fractions to decimals

Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of

### Algebra Cheat Sheets

Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

### Math Circle Beginners Group October 18, 2015

Math Circle Beginners Group October 18, 2015 Warm-up problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd

### REVIEW SHEETS INTRODUCTORY PHYSICAL SCIENCE MATH 52

REVIEW SHEETS INTRODUCTORY PHYSICAL SCIENCE MATH 52 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts which are taught in the specified math course.

### ACCUPLACER Arithmetic & Elementary Algebra Study Guide

ACCUPLACER Arithmetic & Elementary Algebra Study Guide Acknowledgments We would like to thank Aims Community College for allowing us to use their ACCUPLACER Study Guides as well as Aims Community College

### Integers, I, is a set of numbers that include positive and negative numbers and zero.

Grade 9 Math Unit 3: Rational Numbers Section 3.1: What is a Rational Number? Integers, I, is a set of numbers that include positive and negative numbers and zero. Imagine a number line These numbers are

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### UNIT 5 VOCABULARY: POLYNOMIALS

2º ESO Bilingüe Page 1 UNIT 5 VOCABULARY: POLYNOMIALS 1.1. Algebraic Language Algebra is a part of mathematics in which symbols, usually letters of the alphabet, represent numbers. Letters are used to

### Inequalities - Absolute Value Inequalities

3.3 Inequalities - Absolute Value Inequalities Objective: Solve, graph and give interval notation for the solution to inequalities with absolute values. When an inequality has an absolute value we will

### Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

### 1.1 A Modern View of the Universe" Our goals for learning: What is our place in the universe?"

Chapter 1 Our Place in the Universe 1.1 A Modern View of the Universe What is our place in the universe? What is our place in the universe? How did we come to be? How can we know what the universe was

### Binary Number System. 16. Binary Numbers. Base 10 digits: 0 1 2 3 4 5 6 7 8 9. Base 2 digits: 0 1

Binary Number System 1 Base 10 digits: 0 1 2 3 4 5 6 7 8 9 Base 2 digits: 0 1 Recall that in base 10, the digits of a number are just coefficients of powers of the base (10): 417 = 4 * 10 2 + 1 * 10 1

### 3.3 Addition and Subtraction of Rational Numbers

3.3 Addition and Subtraction of Rational Numbers In this section we consider addition and subtraction of both fractions and decimals. We start with addition and subtraction of fractions with the same denominator.

### Principles 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

### Basic numerical skills: FRACTIONS, DECIMALS, PROPORTIONS, RATIOS AND PERCENTAGES

Basic numerical skills: FRACTIONS, DECIMALS, PROPORTIONS, RATIOS AND PERCENTAGES. Introduction (simple) This helpsheet is concerned with the ways that we express quantities that are not whole numbers,

### MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.

1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with

### CAHSEE on Target UC Davis, School and University Partnerships

UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

### Expression. 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

### HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

### Answer 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.

### Stoichiometry Exploring a Student-Friendly Method of Problem Solving

Stoichiometry Exploring a Student-Friendly Method of Problem Solving Stoichiometry comes in two forms: composition and reaction. If the relationship in question is between the quantities of each element

### Linear Equations and Inequalities

Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109 - Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................