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


 Terence Welch
 2 years ago
 Views:
Transcription
1 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 a 0. The form 0 0 is called indeterminate and is considered in later mathematics classes. Note that in 6 0 the eponent 0 applied onl to. Zero and Negative Eponents and Scientific Notation In Section., we eamined properties of eponents, but all of the eponents were positive integers. In this section, we look at zero and negative eponents. First we etend the quotient rule so that we can define an eponent of zero. Recall that, in the quotient rule, to divide two epressions that have the same base, we keep the base and subtract the eponents. m a a n a mn Now, suppose that we allow m to equal n. We then have But we know that it is also true that m a m a amm a 0 () m a a m () Comparing equations () and (), we see that the following definition is reasonable. The Zero Eponent For an real number a where a 0, The Zero Eponent Use the above definition to simplif each epression. (a) 7 0 (b) (a b ) 0 (c) (d) 0 a 0
2 Section 0. Zero and Negative Eponents and Scientific Notation 667 CHECK YOURSELF Simplif each epression. (a) 0 (b) (m n ) 0 (c) 8s 0 (d) 7t 0 Recall that, in the product rule, to multipl epressions with the same base, keep the base and add the eponents. a m a n a mn Now, what if we allow one of the eponents to be negative and appl the product rule? Suppose, for instance, that m and m. Then so a m a n a a a () a 0 a a Negative Integer Eponents John Wallis (66 70), an English mathematician, was the first to full discuss the meaning of 0, negative, and rational eponents (which we discuss in Section 0.). For an nonzero real number a and whole number n, a n a n and a n is the multiplicative inverse of a n. Eample illustrates this definition. E a m p l e From this point on, to simplif will mean to write the epression with positive eponents onl. Also, we will restrict all variables so that the represent nonzero real numbers. Using Properties of Eponents Simplif the following epressions. (a) (b) 6
3 668 Chapter 0 Radicals and Eponents (c) () ( ) (d) 8 7 CHECK YOURSELF Simplif each of the following epressions. (a) a 0 (b) (c) () (d) Eample illustrates the case where coefficients are involved in an epression with negative eponents. As will be clear, some caution must be used. E a m p l e Using Properties of Eponents Simplif each of the following epressions. (a) Caution The epressions w and (w) are not the same. Do ou see wh?! The eponent applies onl to the variable, and not to the coefficient. (b) w w w (c) (w) (w ) 6 w CHECK YOURSELF Simplif each of the following epressions. (a) w (b) 0 (c) () (d) t
4 Section 0. Zero and Negative Eponents and Scientific Notation 669 Suppose that a variable with a negative eponent appears in the denominator of an epression. Our previous definition can be used to write a comple fraction that can then be simplified. For instance, a a a a Negative eponent in denominator Positive eponent in numerator To divide, we invert and multipl. To avoid the intermediate steps, we can write that, in general, For an nonzero real number a and integer n, a n a n E a m p l e Using Properties of Eponents Simplif each of the following epressions. (a) (b) (c) The eponent applies onl to, not to. (d) a b b a CHECK YOURSELF Simplif each of the following epressions. (a) (b) (c) a c (d) d 7 To review these properties, return to Section.. The product and quotient rules for eponents appl to epressions that involve an integral eponent positive, negative, or 0. Eample illustrates this concept.
5 670 Chapter 0 Radicals and Eponents E a m p l e Using Properties of Eponents Simplif each of the following epressions, and write the result, using positive eponents onl. (a) 7 (7) (b) m m m () m m m () (c) (7) 9 Add the eponents b the product rule. Subtract the eponents b the quotient rule. We appl first the product rule and then the quotient rule. Note that m in the numerator becomes m in the denominator, and m in the denominator becomes m in the numerator. We then simplif as before. In simplifing epressions involving negative eponents, there are often alternate approaches. For instance, in Eample, part b, we could have made use of our earlier work to write m m m m m m m CHECK YOURSELF Simplif each of the following epressions. (a) 9 (b) 7 a (c) a a The properties of eponents can be etended to include negative eponents. One of these properties, the quotientpower rule, is particularl useful when rational epressions are raised to a negative power. Let s look at the rule and appl it to negative eponents. QuotientPower Rule a b n a b n n
6 Section 0. Zero and Negative Eponents and Scientific Notation 67 Raising Quotients to a Negative Power a b n a b n n n b a n b a n a 0, b 0 E a m p l e 6 Etending the Properties of Eponents Simplif each epression. (a) s t s t s t 6 6 m (b) n n n m 6 m m 6 n 6 CHECK YOURSELF 6 Simplif each epression. (a) t s (b) E a m p l e 7 Using Properties of Eponents Simplif each of the following epressions. (a) q q q 9 0 (b) ( ) 9 ( )
7 67 Chapter 0 Radicals and Eponents CHECK YOURSELF 7 Simplif each of the following epressions. (a) r (b) a b As ou might epect, more complicated epressions require the use of more than one of the properties, for simplification. Eample 8 illustrates such cases. E a m p l e 8 Using Properties of Eponents Simplif each of the following epressions. ) (a) (a ) ( a a 9 ( a ) a 6 a 6 a 9 a a a 9 a 6(9) a 69 a 6 Appl the power rule to each factor. Appl the product rule. Appl the quotient rule. It ma help to separate the problem into three fractions, one for the coefficients and one for each of the variables. (b) 8 8 () 8 8 Caution! s (c) pr p r s ( p r () s () ) ( p r 6 s ) ( p ) (r 6 ) (s ) 6 p r s 6 p s r Appl the quotient rule inside the parentheses. Appl the rule for a product to a power. Appl the power rule. Be Careful! Another possible first step (and generall an efficient one) is to rewrite an epression b using our earlier definitions. a n a n and a n a n For instance, in Eample 8, part b, we would correctl write 8 8
8 Section 0. Zero and Negative Eponents and Scientific Notation 67 Caution! A common error is to write 8 8 This is not correct. The coefficients should not have been moved along with the factors in. Keep in mind that the negative eponents appl onl to the variables. The coefficients remain where the were in the original epression when the epression is rewritten using this approach. CHECK YOURSELF 8 Simplif each of the following epressions. (a) ( ( ) ( ) (b) a b ) 6a b (c) z z Let us now take a look at an important use of eponents, scientific notation. We begin the discussion with a calculator eercise. On most calculators, if ou multipl. times 000, the displa will read 00 Multipl b 000 a second time. Now ou will see Multipling b 000 a third time will result in the displa This must equal,00,000,000. Consider the following table: , , And multipling b 000 again ields. 09 or. E09. or. E Can ou see what is happening? This is the wa calculators displa ver large numbers. The number on the left is alwas 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 frequentl in science. It is not uncommon in scientific applications of algebra to find ourself working with ver large or ver small numbers. Even in the time of Archimedes (87 B.C.), the stud of such numbers was not unusual. Archimedes estimated that the universe was,000,000,000,000,000 m in diameter, which is the approimate distance light travels in ears. B comparison, Polaris (the North Star) is 680 lightears from the earth. Eample 0 will discuss the idea of lightears.
9 67 Chapter 0 Radicals and Eponents In scientific notation, his estimate for the diameter of the universe would be. 0 6 m In general, we can define scientific notation as follows. Scientific Notation An number written in the form a 0 n where a 0 and n is an integer, is written in scientific notation. E a m p l e 9 Note the pattern for writing a number in scientific notation. The eponent on 0 shows the number of places we must move the decimal point so that the multiplier will be a number between and 0. A positive eponent tells us to move right, while a negative eponent indicates to move left. Note: To convert back to standard or decimal form, the process is simpl reversed. Using Scientific Notation Write each of the following numbers in scientific notation. (a) 0, places The power is. (b) 88,000, places The power is 7. (c) 0,000, places (d),000,000, places (e) places (f) places If the decimal point is to be moved to the left, the eponent will be negative. CHECK YOURSELF 9 Write in scientific notation. (a),000,000,000,000,000 (b) (c),600,000 (d)
10 Section 0. Zero and Negative Eponents and Scientific Notation 67 E a m p l e 0 An Application of Scientific Notation (a) Light travels at a speed of meters per second (m/s). There are approimatel. 0 7 s in a ear. How far does light travel in a ear? We multipl the distance traveled in s b the number of seconds in a ear. This ields Note that We divide the distance (in meters) b the number of meters in lightear. ( )(. 0 7 ) (.0.)( ) Multipl the coefficients, and add the eponents For our purposes we round the distance light travels in ear to 0 6 m. This unit is called a lightear, and it is used to measure astronomical distances. (b) The distance from earth to the star Spica (in Virgo) is. 0 8 m. How man lightears is Spica from earth? lightears CHECK YOURSELF 0 The farthest object that can be seen with the unaided ee is the Andromeda gala. This gala is. 0 m from earth. What is this distance in lightears? CHECK YOURSELF ANSWERS. (a) ; (b) ; (c) 8; (d) 7.. (a) a 0 ; (b) ; (c) ; (d) (a) w ; (b) 0 ; (c) ; 6 (d) t.. (a) ; (b) 7; (c) a ; (d) d 7. c. (a) ; (b) ; (c) a 9 s. 6. (a) ; 7t 6 (b) (a) 9 b r 8; (b) a. 8. (a) 8 ; (b) ; a b (c) z. 9. (a). 0 7 ; (b) ; (c) ; (d) ,00,000 lightears.
11 E e r c i s e s In Eercises to, simplif each epression () 6. () 7. () 8. () (). () a 6. w 7. z 0 8. b 8 In Eercises to, use the properties of eponents to simplif epressions.... a 9 a 6 6. w w 7. z z 8 8. b 7 b 9. a a
12 Section 0. Zero and Negative Eponents and Scientific Notation w. 6. 9p 0 7. a r 0 s 7. a 7 b 8 c. p q 7 r.. 6. b 8 6. b p q b a In Eercises to 8, use the properties of eponents to simplif the following.. ( ). (w ) 6. ( )( ) 6. (p )(p ) 7. (a )(a )(a ) 8. ( )()( ) 9. ( )( ) ( ) 0 0. (r ) (r s)(s ). (ab c)(a ) (b ) (c ). (p qr )(p )(q ) (r ) 0. ( ). ( ). (b ) 6. (a 0 b ) 7. ( ) 8. (p q ) 9. ( ) 0. ( ). ( 0 ). a b ( ) ( ) , ( ) ( ) 8. ( ) ( ) z 6. z a w z 6 7. w 7. 0 b a z z In Eercises 9 to 90, simplif each epression. 9. ( ) ( ) 60. ( ) ( ) ( ) 6. ( ) ( ) 6. ( ) ( ) 0 6. ( z) (z ) 8 ( 6 z) 6. ( z ) 0 ( z) ( z ) 6. ( )( ) 66. (a ) (a 0 ) 67. (w ) (w ) 68. ( ) ( ) (7 )( 6 ) 7. w z 6 9 w ( )( ) z 7. (a b )(a b 0 ) 7. ( )( ) z 78. z 6 z z
13 678 Chapter 0 Radicals and Eponents z n 8. n n n 87. n m z z n 8. n n 8. n n 8. n n 86. n 87. ( n ) n 88. ( n ) n n 89. n n n 90. n n In Eercises 9 to 9, epress each number in scientific notation. 9. The distance from the earth to the sun: 9,000,000 mi. 9. The diameter of a grain of sand: m. 9. The diameter of the sun: 0,000,000,000 cm. 9. The number of molecules in. L of a gas: 60,000,000,000,000,000,000,000 (Avogadro s number) 9. The mass of the sun is approimatel kg. If this were written in standard or decimal form, how man 0s would follow the digit 8? 96. Archimedes estimated the universe to be. 0 9 millimeters (mm) in diameter. If this number were written in standard or decimal form, how man 0s would follow the digit? In Eercises 97 to 00, write each epression in standard notation In Eercises 0 to 0, write each of the following in scientific notation In Eercises 0 to 08, compute the epressions using scientific notation, and write our answer in that form. 0. ( 0 )( 0 ) 06. (. 0 6 )( 0 ) In Eercises 09 to, perform the indicated calculations. Write our result in scientific notation ( 0 )( 0 ) 0. (. 0 7 )( 0 ) (. 0 )(6 0 ) (. 0 8 )( 0 6 ) (6 0 )(. 0 8 ) ( )( 0 )
14 Section 0. Zero and Negative Eponents and Scientific Notation ears 6. 0 ears ; L Megrez, the nearest of the Big Dipper stars, is m from earth. Approimatel how long does it take light, traveling at 0 6 m/ear, to travel from Megrez to earth? 6. Alkaid, the most distant star in the Big Dipper, is. 0 8 m from earth. Approimatel how long does it take light to travel from Alkaid to earth? 7. The number of liters of water on earth is,00 followed b 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. billion. 8. If there are. 0 9 people on earth and there is enough freshwater to provide each person with L, how much freshwater is on earth? 9. The United States uses an average of L of water per person each ear. The United States has. 0 8 people. How man liters of water does the United States use each ear? 0. Can (a b) be written as a b using the properties of eponents? If not, wh not? Eplain. b. Write a short description of the difference between (),,(), and. Are an of these equal?. If n 0, which of the following epressions are negative? n, n,(n),(n), n If n 0, which of these epressions are negative? Eplain what effect a negative in the eponent has on the sign of the result when an eponential epression is simplified.. Take the best offer. You are offered a 8da job in which ou have a choice of two different pa arrangements. Plan offers a flat $,000,000 at the end of the 8th da on the job. Plan offers the first da, the second da, the third da, and so on, with the amount doubling each da. Make a table to decide which offer is the best. Write a formula for the amount ou make on the nth da and a formula for the total after n das. Which pa arrangement should ou take? Wh?
Negative Exponents and Scientific Notation
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
More informationNegative 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
More informationTHE POWER RULES. Raising an Exponential Expression to a Power
8 (5) Chapter 5 Eponents and Polnomials 5. THE POWER RULES In this section Raising an Eponential Epression to a Power Raising a Product to a Power Raising a Quotient to a Power Variable Eponents Summar
More informationSimplification of Rational Expressions and Functions
7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work
More informationNegative 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
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More informationRational Exponents. Given that extension, suppose that. Squaring both sides of the equation yields. a 2 (4 1/2 ) 2 a 2 4 (1/2)(2) a a 2 4 (2)
SECTION 0. Rational Exponents 0. OBJECTIVES. Define rational exponents. Simplify expressions with rational exponents. Estimate the value of an expression using a scientific calculator. Write expressions
More informationExponents. Learning Objectives 41
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
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 informationA.1 Radicals and Rational Exponents
APPENDIX A. Radicals and Rational Eponents 779 Appendies Overview This section contains a review of some basic algebraic skills. (You should read Section P. before reading this appendi.) Radical and rational
More informationSimplifying Exponential Expressions
Simplifying Eponential Epressions Eponential Notation Base Eponent Base raised to an eponent Eample: What is the base and eponent of the following epression? 7 is the base 7 is the eponent Goal To write
More informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More informationa > 0 parabola opens a < 0 parabola opens
Objective 8 Quadratic Functions The simplest quadratic function is f() = 2. Objective 8b Quadratic Functions in (h, k) form Appling all of Obj 4 (reflections and translations) to the function. f() = a(
More informationA 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
More informationThe wavelength of infrared light is meters. The digits 3 and 7 are important but all the zeros are just place holders.
Section 6 2A: A common use of positive and negative exponents is writing numbers in scientific notation. In astronomy, the distance between 2 objects can be very large and the numbers often contain many
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 informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationeday Lessons Mathematics Grade 8 Student Name:
eday Lessons Mathematics Grade 8 Student Name: Common Core State Standards Expressions and Equations Work with radicals and integer exponents. 3. Use numbers expressed in the form of a single digit times
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) 0 C)
Sample/Practice Final Eam MAT 09 Beginning Algebra Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the epression. 1) 2  [  ( 38)]
More informationSolutions 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
More informationMultiplying and Dividing Algebraic Fractions
. Multiplying and Dividing Algebraic Fractions. OBJECTIVES. Write the product of two algebraic fractions in simplest form. Write the quotient of two algebraic fractions in simplest form. Simplify a comple
More informationName Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE
Name Date Block Know how to Algebra 1 Laws of Eponents/Polynomials Test STUDY GUIDE Evaluate epressions with eponents using the laws of eponents: o a m a n = a m+n : Add eponents when multiplying powers
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationx 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac
Solving Quadratic Equations a b c 0, a 0 Methods for solving: 1. B factoring. A. First, put the equation in standard form. B. Then factor the left side C. Set each factor 0 D. Solve each equation. B square
More informationSimplification Problems to Prepare for Calculus
Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.
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 informationExponential and Logarithmic Functions
Chapter 3 Eponential and Logarithmic Functions Section 3.1 Eponential Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph eponential functions. Course
More informationOperations on Decimals
Operations on Decimals Addition and subtraction of decimals To add decimals, write the numbers so that the decimal points are on a vertical line. Add as you would with whole numbers. Then write the decimal
More informationFind each missing factor = 5 2 (j) 6. New Vocabulary rationalize the denominator EXAMPLE. 1 Multiply. Simplify if possible. b. " 3 25?
7. Plan 7 Multipling and Dividing Radical Expressions Objectives To multipl radical expressions To divide radical expressions Examples Multipling Radicals Simplifing Radical Expressions Multipling Radical
More informationRational Expressions  Complex Fractions
7. Rational Epressions  Comple Fractions Objective: Simplify comple fractions by multiplying each term by the least common denominator. Comple fractions have fractions in either the numerator, or denominator,
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
More informationSubstitute 4 for x in the function, Simplify.
Page 1 of 19 Review of Eponential and Logarithmic Functions An eponential function is a function in the form of f ( ) = for a fied ase, where > 0 and 1. is called the ase of the eponential function. The
More information13 Absolute Value in Equations and Inequalities
SECTION 1 3 Absolute Value in Equations and Inequalities 103 13 Absolute Value in Equations and Inequalities Z Relating Absolute Value and Distance Z Solving Absolute Value Equations and Inequalities
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More informationAre You Ready? Simplify Radical Expressions
SKILL Are You Read? Simplif Radical Epressions Teaching Skill Objective Simplif radical epressions. Review with students the definition of simplest form. Ask: Is written in simplest form? (No) Wh or wh
More informationSummer Review For Students Entering Algebra 2
Summer Review For Students Entering Algebra Board of Education of Howard Count Frank Aquino Chairman Ellen Flnn Giles Vice Chairman Larr Cohen Allen Der Sandra H. French Patricia S. Gordon Janet Siddiqui
More informationSection A3 Polynomials: Factoring APPLICATIONS. A22 Appendix A A BASIC ALGEBRA REVIEW
A Appendi A A BASIC ALGEBRA REVIEW C In Problems 53 56, perform the indicated operations and simplify. 53. ( ) 3 ( ) 3( ) 4 54. ( ) 3 ( ) 3( ) 7 55. 3{[ ( )] ( )( 3)} 56. {( 3)( ) [3 ( )]} 57. Show by
More informationPolynomial and Rational Functions
Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important
More informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real
More informationGraphing Nonlinear Systems
10.4 Graphing Nonlinear Sstems 10.4 OBJECTIVES 1. Graph a sstem of nonlinear equations 2. Find ordered pairs associated with the solution set of a nonlinear sstem 3. Graph a sstem of nonlinear inequalities
More informationMATH Fundamental Mathematics II.
MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/funmath2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics
More informationQuadratic Equations and Functions
Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In
More information12.3 Inverse Matrices
2.3 Inverse Matrices Two matrices A A are called inverses if AA I A A I where I denotes the identit matrix of the appropriate size. For example, the matrices A 3 7 2 5 A 5 7 2 3 If we think of the identit
More informationThe Distance Formula and the Circle
10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,
More information3.1 Solving Quadratic Equations by Taking Square Roots
COMMON CORE 816 1 1 10 8 6 0 y Locker LESSON.1 Solving Quadratic Equations by Taking Square Roots Name Class Date.1 Solving Quadratic Equations by Taking Square Roots Essential Question: What is an imaginary
More informationSolving Absolute Value Equations and Inequalities Graphically
4.5 Solving Absolute Value Equations and Inequalities Graphicall 4.5 OBJECTIVES 1. Draw the graph of an absolute value function 2. Solve an absolute value equation graphicall 3. Solve an absolute value
More informationRational Exponents and Radical Functions
Rational Eponents and Radical Functions.1 nth Roots and Rational Eponents. Properties of Rational Eponents and Radicals. Graphing Radical Functions. Solving Radical Equations and Inequalities. Performing
More informationax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as
More informationSECTION 51 Exponential Functions
354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational
More informationRational Expressions and Rational Equations
Rational Epressions and Rational Equations 6 6. Rational Epressions and Rational Functions 6. Multiplication and Division of Rational Epressions 6. Addition and Subtraction of Rational Epressions 6.4 Comple
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 informationPolynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will
More informationSect Exponents: Multiplying and Dividing Common Bases
40 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 informationReteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button.
Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.
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 informationEssential Question: What are two ways to solve an absolute value inequality? A2.6.F Solve absolute value linear inequalities.
Locker LESSON.3 Solving Absolute Value Inequalities Name Class Date.3 Solving Absolute Value Inequalities Teas Math Standards The student is epected to: A.6.F Essential Question: What are two was to solve
More information2.2 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,
More informationSECTION 14 Absolute Value in Equations and Inequalities
14 Absolute Value in Equations and Inequalities 37 SECTION 14 Absolute Value in Equations and Inequalities Absolute Value and Distance Absolute Value in Equations and Inequalities Absolute Value and
More informationSession 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
More information1.6 Graphs of Functions
.6 Graphs of Functions 9.6 Graphs of Functions In Section. we defined a function as a special tpe of relation; one in which each coordinate was matched with onl one coordinate. We spent most of our time
More informationEquations Involving Fractions
. Equations Involving Fractions. OBJECTIVES. Determine the ecluded values for the variables of an algebraic fraction. Solve a fractional equation. Solve a proportion for an unknown NOTE The resulting equation
More informationZeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.
_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial
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 informationIf (a)(b) 5 0, then a 5 0 or b 5 0.
chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic
More informationIntegration of Rational Expressions by Partial Fractions
Integration of Rational Epressions by Partial Fractions INTRODUTION: We start with a few definitions rational epression is formed when a polynomial is divided by another polynomial In a proper rational
More informationSection 0.3 Power and exponential functions
Section 0.3 Power and eponential functions (5/6/07) Overview: As we will see in later chapters, man mathematical models use power functions = n and eponential functions =. The definitions and asic properties
More informationIntroduction to the Practice Exams
Introduction to the Practice Eams The math placement eam determines what math course you will start with at North Hennepin Community College. The placement eam starts with a 1 question elementary algebra
More information14.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes. Learning Style
The Area Bounded b a Curve 14.3 Introduction One of the important applications of integration is to find the area bounded b a curve. Often such an area can have a phsical significance like the work done
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More informationCSE 1400 Applied Discrete Mathematics Conversions Between Number Systems
CSE 400 Applied Discrete Mathematics Conversions Between Number Systems Department of Computer Sciences College of Engineering Florida Tech Fall 20 Conversion Algorithms: Decimal to Another Base Conversion
More informationAlgebra ConceptReadiness Test, Form A
Algebra ConceptReadiness Test, Form A Concept : The Distributive Property Study the concept, and then answer the test questions on the net page. You can use the distributive property to simplify an epression
More informationExponential Functions
MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 7 Eponential Functions e 20 1 e 10 (0, 1) 4 2 0 2 4 MATHS LEARNING CENTRE Level 3,
More information3.1. 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
More informationHigher. Functions and Graphs. Functions and Graphs 18
hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationQ (x 1, y 1 ) m = y 1 y 0
. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine
More information6.3. section. Building Up the Denominator. To convert the fraction 2 3 factor 21 as 21 3 7. Because 2 3
0 (618) Chapter 6 Rational Epressions GETTING MORE INVOLVED 7. Discussion. Evaluate each epression. a) Onehalf of 1 b) Onethird of c) Onehalf of d) Onehalf of 1 a) b) c) d) 8 7. Eploration. Let R
More information1.2 GRAPHS OF EQUATIONS
000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the  and intercepts of graphs of equations. Write the standard forms of equations of
More informationHIRES STILL TO BE SUPPLIED
1 MRE GRAPHS AND EQUATINS HIRES STILL T BE SUPPLIED Differentshaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object
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 informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationSkill Builders. (Extra Practice) Volume I
Skill Builders (Etra Practice) Volume I 1. Factoring Out Monomial Terms. Laws of Eponents 3. Function Notation 4. Properties of Lines 5. Multiplying Binomials 6. Special Triangles 7. Simplifying and Combining
More informationSolving Equations, Formulas, and Proportions
Solving Equations, Formulas, and Proportions Section : Introduction One of the basic goals of algebra is solving equations. An equation is a mathematical statement in which two epressions equal one another.
More informationContents. How You May Use This Resource Guide
Contents How You Ma Use This Resource Guide ii 9 Fractional and Quadratic Equations 1 Worksheet 9.1: Similar Figures.......................... 5 Worksheet 9.: Stretch of a Spring........................
More informationSection C Non Linear Graphs
1 of 8 Section C Non Linear Graphs Graphic Calculators will be useful for this topic of 8 Cop into our notes Some words to learn Plot a graph: Draw graph b plotting points Sketch/Draw a graph: Do not plot,
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More informationGRE MATH REVIEW #5. 1. Variable: A letter that represents an unknown number.
GRE MATH REVIEW #5 Eponents and Radicals Many numbers can be epressed as the product of a number multiplied by itself a number of times. For eample, 16 can be epressed as. Another way to write this is
More informationFree PreAlgebra Lesson 55! page 1
Free PreAlgebra 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 informationSTUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS
STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an
More informationLINEAR 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
More informationMathematical goals. Starting points. Materials required. Time needed
Level A7 of challenge: C A7 Interpreting functions, graphs and tables tables Mathematical goals Starting points Materials required Time needed To enable learners to understand: the relationship between
More informationThe Graph of a Linear Equation
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
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 yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationMore Equations and Inequalities
Section. Sets of Numbers and Interval Notation 9 More Equations and Inequalities 9 9. Compound Inequalities 9. Polnomial and Rational Inequalities 9. Absolute Value Equations 9. Absolute Value Inequalities
More information10.2 The Unit Circle: Cosine and Sine
0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on
More informationLinear Equations in Two Variables
Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations
More informationRight Triangles and SOHCAHTOA: Finding the Length of a Side Given One Side and One Angle
Right Triangles and SOHCAHTOA: Finding the Length of a Side Given One Side and One Angle Preliminar Information: is an acronm to represent the following three trigonometric ratios or formulas: opposite
More information