Mathematics
Powers and Exponents
Powers and exponents in mathematics refer to the shorthand notation for repeated multiplication. A power is represented as a base number raised to an exponent, indicating the number of times the base is multiplied by itself. Exponents can be used to simplify and solve complex mathematical expressions, making calculations more efficient and manageable.
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eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
1 MODULE Exponents and logarithms 1.1 Exponents On completion of this topic, you should be able to: 1.1.1 Identify the sign, coefficient, base and exponent of a power 1.1.2 Give the laws of exponents 1.1.3 Apply the laws for exponents in simplifying algebraic expressions. 1.2 Logarithms On completion of this topic, you should be able to: 1.2.1 Define a logarithm 1.2.2 Give the following three laws of logarithms: i) log a xy = log a x + log a y ii) log a x y = log a x – log a y iii) log a x n = n log a x , where a = e , 2 and 10 1.2.3 Simplify simple logarithmic expressions without the use of a calculator 1.2.4 Simplify logarithms using a calculator. 2 Module 1 • Exponents and logarithms 1.1 Exponents (indices) Introduction An exponent (index) is a mathematical way of representing a very large number or a very small number in a format that is user-friendly. It is more or less a shorthand notation for multiplying the same number by itself several times, for example: 2 × 2 × 2 × 2 × 2 = 2 5 = 32 Scientific notation is used to express numbers in a manageable form, for example: 11 000 000 000 = 1,1 × 10 10 and 0,000011 = 1,1 × 10 –5 Every computer programmer has to learn exponents. For example, the term nanometre means 10 –9 metre. In the computer world we work with megabytes ( mega means 10 6 or one million), gigabytes ( giga : 10 9 ) and terabytes ( tera : 10 12 ). Exponents are also used in everyday life when we speak about square metres (m 2 ), square kilometres (km 2 ) or any other area units, or when we speak about cubic metres (m 3 ), cubic centimetres (cm 3 ) or any other volume units, for example: to lay carpets in your room you will need 3 × 4 m 2 carpeting. Scientist use exponents to measure the size of an earthquake. 1.1.1 The sign, coefficient, base and exponent of a power Definitions A coefficient is the numerical part of the expression.
eBook - PDF- Alan Tussy, Diane Koenig(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. CHAPTER 10 • Exponents and Polynomials 932 OBJECTIVE 1 Identify bases and exponents. Recall that an exponent indicates repeated multiplication. It indicates how many times the base is used as a factor. For example, 3 5 represents the product of five 3’s. Exponent 5 factors of 3 3 5 5 3 ? 3 ? 3 ? 3 ? 3 Base In general, we have the following definition. Natural-Number Exponents A natural-number* exponent tells how many times its base is to be used as a factor. For any number x and any natural number n, n factors of x x n 5 x ? x ? x ? p ? x *The set of natural numbers is {1, 2, 3, 4, 5, … }. Expressions of the form x n are called exponential expressions. The base of an exponential expression can be a number, a variable, or a combination of numbers and variables. Some examples are: 10 5 5 10 ? 10 ? 10 ? 10 ? 10 The base is 10. The exponent is 5. Read as “10 to the fifth power.” y 2 5 y ? y The base is y. The exponent is 2. Read as “y squared.” (22s) 3 5 (22s)(22s)(22s) The base is 22s. The exponent is 3. Read as “negative 2s raised to the third power” or “negative 2s cubed.” 28 4 5 2(8 ? 8 ? 8 ? 8) Since the 2 sign is not written within parentheses, the base is 8. The exponent is 4. Read as “the opposite (or the negative) of 8 to the fourth power.” When an exponent is 1, it is usually not written. For example, 4 5 4 1 and x 5 x 1 . OBJECTIVES 1 Identify bases and exponents. 2 Multiply exponential expressions that have like bases. 3 Raise exponential expressions to a power. 4 Find powers of products. SECTION 10.1 Multiplication Rules for Exponents In this section, we will use the definition of exponent to develop some rules for simplifying expressions that contain exponents.
eBook - PDF- Charles P. McKeague, Kate Duffy Pawlik(Authors)
- 2014(Publication Date)
- XYZ Textbooks(Publisher)
It seems simple, but it is up to you to see that you maintain the skills. If you intend to take more classes in mathematics and want to ensure your suc- cess then you can work toward this goal: Become a student who can learn math- ematics on his or her own. Most people who have degrees in mathematics were students who could learn mathematics on their own. This doesn’t mean that you always have to learn it on your own; it simply means that if you have to, you can. When you reach this goal, you'll be in control of your success in any math class you take. © Aga & Miko Materne/iStockPhoto 10.1 10.1 Multiplication with Exponents 599 Multiplication with Exponents Recall that an exponent is a number written just above and to the right of another number, which is called the base. In the expression 5 2 , for example, the exponent is 2 and the base is 5. The expression 5 2 is read “5 to the second power” or “5 squared.” The meaning of the expression is 5 2 = 5 ⋅ 5 = 25 In the expression 5 3 , the exponent is 3 and the base is 5. The expression 5 3 is read “5 to the third power” or “5 cubed.” The meaning of the expression is 5 3 = 5 ⋅ 5 ⋅ 5 = 125 Here are some further examples. Write each expression as a single number. a. 4 3 b. −3 4 c. (−2) 5 d. − 3 __ 4 2 Solution a. 4 3 = 4 ⋅ 4 ⋅ 4 = 16 ⋅ 4 = 64 Exponent 3, base 4 b. −3 4 = −3 ⋅ 3 ⋅ 3 ⋅ 3 = −81 Exponent 4, base 3 c. (−2) 5 = (−2)(−2)(−2)(−2)(−2) = −32 Exponent 5, base −2 d. − 3 __ 4 2 = − 3 __ 4 − 3 __ 4 = 9 __ 16 Exponent 2, base − 3 __ 4 Question In what way are (−5) 2 and −5 2 different? Answer In the first case, the base is −5. In the second case, the base is 5. The answer to the first is 25. The answer to the second is −25. Can you tell why? Would there be a difference in the answers if the exponent in each case were changed to 3? We can simplify our work with exponents by developing some properties of exponents.
eBook - PDF- Alan Tussy, R. Gustafson(Authors)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
Take notes as you watch the explanations. Now Try This SECTION 5.1 Rules for Exponents Identify bases and exponents. Multiply exponential expressions that have like bases. Divide exponential expressions that have like bases. Raise exponential expressions to a power. Find powers of products and quotients. OBJECTIVES 1. Evaluate: a. b. 2. Evaluate: a. 2 6 b. 3. Simplify: 4. Evaluate: 4 3 4 2 x x x x x x x 2 6 5 5 5 5 5 5 ARE YOU READY? The following problems review some basic skills that are needed when working with exponents. In this section, we will use the definition of exponent to develop some rules for simplifying expressions that contain exponents. Identify Bases and Exponents. Recall that an exponent indicates repeated multiplication. It indicates how many times the base is used as a factor. For example, represents the product of five 3’s. Exponent 5 factors of 3 Base In general, we have the following definition. 3 5 3 3 3 3 3 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ 3 5 A natural-number exponent tells how many times its base is to be used as a factor. For any number and any natural number , factors of x n x x x p x ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ x n n x Natural-Number Exponents Copyright 201 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Expressions of the form are called exponential expressions. The base of an exponential expression can be a number, a variable, or a combination of numbers and variables. Some examples are: The base is 10. The exponent is 5. Read as “10 to the fifth power” or simply as “10 to the fifth.” The base is . The exponent is 2. Read as “ squared.” The base is .
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 6 Exponentiation Exponentiation is a mathematical operation, written as a n , involving two numbers, the base a and the exponent n . When n is a positive integer, exponentiation corresponds to repeated multiplication; in other words, a product of n factors of a : just as multiplication by a positive integer corresponds to repeated addition: The exponent is usually shown as a superscript to the right of the base. The expon-entiation a n can be read as: a raised to the n -th power , a raised to the power [of] n , or possibly a raised to the exponent [of] n , or more briefly as a to the n . Some exponents have their own pronunciation: for example, a 2 is usually read as a squared and a 3 as a cubed . The power a n can be defined also when n is a negative integer, for nonzero a . No natural extension to all real a and n exists, but when the base a is a positive real number, a n can be defined for all real and even complex exponents n via the exponential function e z . Trigonometric functions can be expressed in terms of complex exponentiation. Exponentiation where the exponent is a matrix is used for solving systems of linear differential equations. Exponentiation is used pervasively in many other fields as well, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography. ________________________ WORLD TECHNOLOGIES ________________________ Graphs of y = a x for various bases a : base 10 ( green ), base e ( red ), base 2 ( blue ), and base ½ ( cyan ). Each curve passes through the point (0,1) because any nonzero number raised to the power 0 is 1. At x =1, the y -value equals the base because any number raised to the power 1 is itself.
eBook - PDFAlgebra
A Combined Course 2E
- Charles P. McKeague(Author)
- 2018(Publication Date)
- XYZ Textbooks(Publisher)
GETTING READY FOR CLASS After reading through the preceding section, respond in your own words and in complete sentences. A. What is an exponent? B. In symbols, show that any number other than zero raised to the 0 power is 1. C. Briefly list the steps of the order of operations. D. Find the value of the expression 3(4t − 2) if t = 2. Answers 12. 16 13. 0, 18, 0 12. Find the value of 5x + 2y − 10 when x is 4 and y is 3. 13. Evaluate (x + 3) 2 , x 2 + 9, and x 2 + 6x + 9 when x is −3. a (a + 4) 2 a 2 + 16 a 2 + 8a + 16 −2 (−2 + 4) 2 = 4 (−2) 2 + 16 = 20 (−2) 2 + 8(−2) + 16 = 4 0 (0 + 4) 2 = 16 0 2 + 16 = 16 0 2 + 8(0) + 16 = 16 3 (3 + 4) 2 = 49 3 2 + 16 = 25 3 2 + 8(3) + 16 = 49 82 Chapter 1 Real Numbers and Algebraic Expressions E X E R C I S E S E T 1.2 VOCABULARY REVIEW Choose the correct words to fill in the blanks below. zero first base product difference factor sum exponent quotient 1. For the exponential expression 5 3 , the 5 is called the and the 3 is called the . 2. An exponent is a number that indicates how many times the base is used as a . 3. Any number raised to the power is the number itself. 4. Any non-zero number raised to the power is 1. 5. Twice the of 6 and 1 is written in symbols as 2(6 + 1). 6. 4 added to 5 times the of 3 and 2 is written in symbols as 5(3 − 2) + 4. 7. 7 subtracted from the of 4 and 3 is written in symbols as 4 ∙ 3 − 7. 8. The of 8 and 2 plus 9 is written in symbols as 8 ÷ 2 + 9. The following is a list of steps to perform operations when evaluating mathematical expressions. Number the steps in the correct order. Do all multiplications and divisions in order, left to right. Simplify any numbers with exponents. Perform operations inside the grouping symbols, or above and below the fraction bar. Do all additions and subtractions, in order left to right. A For each of the following expressions, name the base and the exponent. 1. 4 5 2. 5 4 3. 3 6 4. 6 3 5. 8 2 6. 2 8 7. 9 1 8. 1 9 9.
eBook - PDFIntroductory Algebra
Concepts and Graphs 2E
- Charles P. McKeague(Author)
- 2020(Publication Date)
- XYZ Textbooks(Publisher)
The expression 5 2 is read “5 to the second power” or “5 squared.” The meaning of the expression is 5 2 = 5 ⋅ 5 = 25 In the expression 5 3 , the exponent is 3 and the base is 5. The expression 5 3 is read “5 to the third power” or “5 cubed.” The meaning of the expression is 5 3 = 5 ⋅ 5 ⋅ 5 = 125 A Simplifying with Exponents Here are some further examples. Write each expression as a single number. a. 4 3 b. −3 4 c. (−2) 5 d. − 3 __ 4 2 SOLUTION a. 4 3 = 4 ⋅ 4 ⋅ 4 = 16 ⋅ 4 = 64 Exponent 3, base 4 b. −3 4 = −3 ⋅ 3 ⋅ 3 ⋅ 3 = −81 Exponent 4, base 3 c. (−2) 5 = (−2)(−2)(−2)(−2)(−2) = −32 Exponent 5, base −2 d. − 3 __ 4 2 = − 3 __ 4 − 3 __ 4 = 9 __ 16 Exponent 2, base − 3 __ 4 Question: In what way are (−5) 2 and −5 2 different? Answer: In the first case, the base is −5. In the second case, the base is 5. The answer to the first is 25. The answer to the second is −25. Can you tell why? Would there be a difference in the answers if the exponent in each case were changed to 3? We can simplify our work with exponents by developing some properties of exponents. We want to list the things we know are true about exponents and then use these properties to simplify expressions that contain exponents. B Properties of Exponents The first property of exponents applies to products with the same base. We can use the definition of exponents, as indicating repeated multiplication, to simplify expressions like 7 4 ⋅ 7 2 . 7 4 ⋅ 7 2 = (7 ⋅ 7 ⋅ 7 ⋅ 7)(7 ⋅ 7) = (7 ⋅ 7 ⋅ 7 ⋅ 7 ⋅ 7 ⋅ 7) = 7 6 Notice: 4 + 2 = 6 As you can see, multiplication with the same base resulted in addition of expo- nents. We can summarize this result with the following property. If a is any real number and r and s are integers, then a r ⋅ a s = a r+s In words: To multiply two expressions with the same base, add exponents and use the common base. PROPERTY: PROPERTY 1 FOR EXPONENTS EXAMPLE 1 OBJECTIVES A Simplify basic expressions containing exponents.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2017(Publication Date)
- XYZ Textbooks(Publisher)
For example, people making this mistake would get 2 x 15 or (2 x ) 15 as the result in Example 2b. To avoid this mistake, you must be sure you understand the meaning of Property 1 exactly as it is written. If a is any real number and r and s are integers, then ( a r ) s = a r ⋅ s In words: A power raised to another power is the base raised to the product of the powers. PROPERTY Property 2 for Exponents EXAMPLE 3 5.1 Multiplication with Exponents and Scientific Notation 313 The third property of exponents applies to expressions in which the product of two or more numbers or variables is raised to a power. Let’s look at how the expression (2 x ) 3 can be simplified: (2 x ) 3 = (2 x )(2 x )(2 x ) = (2 ⋅ 2 ⋅ 2)( x ⋅ x ⋅ x ) = 2 3 ⋅ x 3 Notice: The exponent 3 distributes over the product 2 x = 8 x 3 We can generalize this result into a third property of exponents. If a and b are any two real numbers and r is an integer, then ( ab ) r = a r b r In words: The power of a product is the product of the powers. PROPERTY Property 3 for Exponents Here are some examples using Property 3 to simplify expressions. Simplify the following expressions: a. (3 y ) 2 b. − 1 __ 4 x 2 y 3 2 c. ( x 2 y 5 ) 3 ( x 4 y ) 2 SOLUTION a. (3 y ) 2 = 3 2 y 2 Property 3 = 9 y 2 b. − 1 __ 4 x 2 y 3 2 = − 1 __ 4 2 ( x 2 ) 2 ( y 3 ) 2 Property 3 = 1 __ 16 x 4 y 6 Property 2 c. ( x 2 y 5 ) 3 ( x 4 y ) 2 = ( x 2 ) 3 ( y 5 ) 3 ⋅ ( x 4 ) 2 y 2 Property 3 = x 6 y 15 ⋅ x 8 y 2 Property 2 = ( x 6 x 8 )( y 15 y 2 ) Commutative and associative properties = x 14 y 17 Property 1 EXAMPLE 4 314 Chapter 5 Exponents and Polynomials Scientific Notation Many branches of science require working with very large numbers. In astronomy, for example, distances commonly are given in light-years. A light-year is the distance light travels in a year. It is approximately 5,880,000,000,000 miles This number is difficult to use in calculations because of the number of zeros it contains.
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