Properties of Exponents

10 Key excerpts on "Properties of Exponents"

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  eBook - PDF

  Elementary Algebra

  • 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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  Intermediate Algebra

  Connecting Concepts through Applications

  • Mark Clark, Cynthia Anfinson(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Recall that a natural number exponent represents repeated multiplication. 2 # 2 # 2 # 2 # 2 # 2 5 2 6 5 64 xxxxx 5 x 5 xxxyy 5 x 3 y 2 3 # 3 # 3 # 3 # 7 # 7 # 7 # 7 # 7 # 7 5 3 4 # 7 6 5 81 # 117649 5 9529569 Exponents allow us to write a long expression in a very compact way. There are two parts to an exponential expression: the base and the exponent. The base is the number or variable being raised to a power. The exponent is the power to which the base is being raised. 5 3 base exponent One of the most common operations done with exponential expressions is to multiply them together. When multiplying any exponential expressions with the same base, the expressions can be combined into one exponential expression. x 7 x 2 5 x x x x x x x # x x 5 x 9 In this example, we see that seven x’s were multiplied by two more x’s, which results in a total of nine x’s multiplied together. Therefore, we can write a final simpler expression, x 9 . Multiplying expressions with the same base leads us to the product rule for exponents. The Product Rule for Exponents x m x n 5 x m 1 n When multiplying exponential expressions with the same base, add the exponents. x 5 x 3 5 x 8 When more than one base is included in an expression or multiplication problem, the associative and commutative properties can be used along with the product rule for exponents to simplify the expression. Rules for Exponents LEARNING OBJECTIVES Use the rules for exponents to simplify expressions. Understand and use negative exponents. Understand and use scientific notation. 3.1 Copyright 2019 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.

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  Introductory Algebra

  Concepts and Graphs 2E

  • Charles P. McKeague(Author)
  • 2020(Publication Date)
  • XYZ Textbooks

    (Publisher)

  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. B Simplify expressions using the first, second, and third Properties of Exponents. C Write numbers using scientific notation. 4.1 VIDEOS 240 CHAPTER 4 Exponents and Polynomials Here is an example using Property 1. Use Property 1 to simplify the following expressions. Leave your answers in terms of exponents: a. 5 3 ⋅ 5 6 b. x 7 ⋅ x 8 c. 3 4 ⋅ 3 8 ⋅ 3 5 SOLUTION a. 5 3 ⋅ 5 6 = 5 3+6 = 5 9 b. x 7 ⋅ x 8 = x 7+8 = x 15 c. 3 4 ⋅ 3 8 ⋅ 3 5 = 3 4+8+5 = 3 17 Another common type of expression involving exponents is one in which an expression containing an exponent is raised to another power. The expression (5 3 ) 2 is an example: (5 3 ) 2 = (5 3 )(5 3 ) = 5 3+3 = 5 6 Notice: 3 ⋅ 2 = 6 This result offers justification for the second property of exponents. 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 prod- uct of the powers. PROPERTY: PROPERTY 2 FOR EXPONENTS Simplify the following expressions: a. (4 5 ) 6 b. (x 3 ) 5 SOLUTION a. (4 5 ) 6 = 4 5⋅6 = 4 30 b. (x 3 ) 5 = x 3⋅5 = x 15 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 (2x) 3 can be simplified: (2x) 3 = (2x)(2x)(2x) = (2 ⋅ 2 ⋅ 2)(x ⋅ x ⋅ x) = 2 3 ⋅ x 3 Notice: The exponent 3 distributes over the product 2x = 8x 3 We can generalize this result into a third property of exponents. EXAMPLE 2 Note: In Example 2, notice that in each case the base in the original problem is the same base that appears in the answer and that it is written only once in the answer.

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  eBook - PDF

  Intermediate Algebra

  Concepts and Graphs 2E

  • Charles P. McKeague(Author)
  • 2019(Publication Date)
  • XYZ Textbooks

    (Publisher)

  3 4 __ 3 4 = 3 4−4 = 3 0 Hence, 3 0 must be the same as 1. Summarizing these results, we have our last property for exponents. EXAMPLE 11 Simplify. a. (2x 2 y 4 ) 0 = 1 b. (2x 2 y 4 ) 1 = 2x 2 y 4 Here are some examples that use many of the Properties of Exponents. There are a number of ways to proceed on problems like these. You should use the method that works best for you. PROPERTY Property 7 for Exponents If a is any real number, then a 1 = a and a 0 = 1 (as long as a ≠ 0) 1.4 Exponents and Scientific Notation 49 EXAMPLE Simplify. 12. (x 3 ) −2 (x 4 ) 5 ________ (x −2 ) 7 = x −6 x 20 _____ x −14 Property 2 = x 14 ____ x −14 Property 1 = x 28 Property 6: x 14−(−14) = x 28 13. 6a 5 b −6 ______ 12a 3 b −9 = 6 __ 12 ⋅ a 5 __ a 3 ⋅ b −6 ___ b −9 Write as separate fractions. = 1 __ 2 a 2 b 3 Property 6 14. (4x −5 y 3 ) 2 _______ (x 4 y −6 ) −3 = 16x −10 y 6 _______ x −12 y 18 Properties 2 and 3 = 16x 2 y −12 Property 6 = 16x 2 ⋅ 1 __ y 12 Property 4 = 16x 2 ____ y 12 Multiplication Scientific Notation Scientific notation is a method for writing very large or very small numbers in a more manageable form. Here is the definition. EXAMPLE 15 Write 376,000 in scientific notation. SOLUTION We must rewrite 376,000 as the product of a number between 1 and 10 and a power of 10. To do so, we move the decimal point five places to the left so that it appears between the 3 and the 7. Then we multiply this number by 10 5 . The number that results has the same value as our original number and is written in scientific notation. Note Example 13 can also be written as a 2 b 3 ___ 2 . Either answer is correct. b DEFINITION scientific notation A number is written in scientific notation if it is written as the product of a number between 1 and 10 and an integer power of 10. A number written in scientific notation has the form n × 10 r where 1 ≤ n < 10 and r = an integer.

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  eBook - PDF

  Elementary Algebra

  • 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 .

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  eBook - PDF

  Prealgebra

  • Alan Tussy, Diane Koenig(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  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. 10.1 • Multiplication Rules for Exponents 935 This example illustrates the following rule for exponents. Power Rule for Exponents To raise an exponential expression to a power, keep the base and multiply the exponents. For any number x and any natural numbers m and n, ( x m ) n 5 x m? n 5 x mn Read as “the quantity of x to the mth power raised to the n th power equals x to the mn th power.” THE LANGUAGE OF ALGEBRA An exponential expression raised to a power, such as (2 3 ) 7 , is also called a power of a power. Simplify: a. (2 3 ) 7 b. [(26) 2 ] 5 c. (z 8 ) 8 Strategy In each case, we want to write an equivalent expression using one base and one exponent. We will use the power rule for exponents to do this. WHY Each expression is a power of a power. Solution a. (2 3 ) 7 5 2 3?7 5 2 21 Keep the base, 2, and multiply the exponents. Since 2 21 is a very large number, we will leave the answer in this form. b. [(26) 2 ] 5 5 (26) 2?5 5 (26) 10 Keep the base, 26, and multiply the exponents. Since (26) 10 is a very large number, we will leave the answer in this form. c. (z 8 ) 8 5 z 8?8 5 z 64 Keep the base, z, and multiply the exponents. EXAMPLE 4 Simplify: a. (4 6 ) 5 b. (y 5 ) 2 Now Try Problems 49, 51, and 53 Self Check 4 Simplify: a. (x 2 x 5 ) 2 b. (z 2 ) 4 (z 3 ) 3 Strategy In each case, we want to write an equivalent expression using one base and one exponent. We will use the product and power rules for exponents to do this.

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  Prealgebra

  • 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.

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  eBook - PDF

  Introductory Algebra

  Concepts with Applications

  • Charles P. McKeague(Author)
  • 2013(Publication Date)
  • XYZ Textbooks

    (Publisher)

  = 1 __ y 2 Negative exponent property The point of this explanation is this: Even though we may not show all the steps when simplifying an expression involving exponents, the result we obtain still can be justified using the Properties of Exponents. We have not introduced any new properties in Example 2; we have just not shown the details of each simplification. EXAMPLE 3 Divide 25a 5 b 3 by 50a 2 b 7 . Solution 25a 5 b 3 _____ 50a 2 b 7 = 25 __ 50 ⋅ a 5 __ a 2 ⋅ b 3 __ b 7 Write as separate fractions. = 1 __ 2 ⋅ a 3 ⋅ 1 __ b 4 Divide coefficients, subtract exponents. = a 3 ___ 2b 4 Write answer as a single fraction. Notice in Example 3 that dividing 25 by 50 results in 1 _ 2 . This is the same result we would obtain if we reduced the fraction 25 __ 50 to lowest terms, and there is no harm in thinking of it that way. Also, notice that the expression b 3 __ b 7 simplifies to 1 _ b 4 by the quotient property of exponents and the negative exponent property, even though we have not shown the steps involved in doing so. Answers 2. a. 3x 3 y b. 5x 3 3. a 2 __ 3b 5 2. Divide. a. 27x 4 y 3 _____ 9xy 2 b. 25x 4 ____ 5x 3. Divide 13a 6 b 2 _____ 39a 4 b 7 . 337 4.3 Operations with Monomials C Multiplication and Division of Numbers Written in Scientific Notation We multiply and divide numbers written in scientific notation using the same steps we used to multiply and divide monomials. EXAMPLE 4 Multiply (4 × 10 7 )(2 × 10 −4 ). Solution Since multiplication is commutative and associative, we can rearrange the order of these numbers and group them as follows: (4 × 10 7 )(2 × 10 −4 ) = (4 × 2)(10 7 × 10 −4 ) = 8 × 10 3 Notice that we add exponents, 7 + (−4) = 3, when we multiply with the same base.

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  eBook - PDF

  Algebra

  A Combined Course 2E

  • Charles P. McKeague(Author)
  • 2018(Publication Date)
  • XYZ Textbooks

    (Publisher)

  = 1 __ y 2 Negative exponent property The point of this explanation is this: Even though we may not show all the steps when simplifying an expression involving exponents, the result we obtain still can be justified using the Properties of Exponents. We have not introduced any new properties in Example 2; we have just not shown the details of each simplification. EXAMPLE 3 Divide 25a 5 b 3 by 50a 2 b 7 . Solution 25a 5 b 3 _____ 50a 2 b 7 = 25 __ 50 ⋅ a 5 __ a 2 ⋅ b 3 __ b 7 Write as separate fractions. = 1 __ 2 ⋅ a 3 ⋅ 1 __ b 4 Divide coefficients, subtract exponents. = a 3 ___ 2b 4 Write answer as a single fraction. Notice in Example 3 that dividing 25 by 50 results in 1 _ 2 . This is the same result we would obtain if we reduced the fraction 25 __ 50 to lowest terms, and there is no harm in thinking of it that way. Also, notice that the expression b 3 __ b 7 simplifies to 1 _ b 4 by the quotient property of exponents and the negative exponent property, even though we have not shown the steps involved in doing so. Answers 2. a. 3x 3 y b. 5x 3 3. a 2 __ 3b 5 2. Divide. a. 27x 4 y 3 _____ 9xy 2 b. 25x 4 ____ 5x 3. Divide 13a 6 b 2 _____ 39a 4 b 7 . 399 5.3 Operations with Monomials C Multiplication and Division of Numbers Written in Scientific Notation We multiply and divide numbers written in scientific notation using the same steps we used to multiply and divide monomials. EXAMPLE 4 Multiply (4 × 10 7 )(2 × 10 −4 ). Solution Since multiplication is commutative and associative, we can rearrange the order of these numbers and group them as follows: (4 × 10 7 )(2 × 10 −4 ) = (4 × 2)(10 7 × 10 −4 ) = 8 × 10 3 Notice that we add exponents, 7 + (−4) = 3, when we multiply with the same base.

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  Mathematics for Information Technology

  • Alfred Basta, Stephan DeLong, Nadine Basta, , Alfred Basta, Stephan DeLong, Nadine Basta(Authors)
  • 2013(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  THE POWER RULE OF LOGARITHMS: For any base a > 0, a ≠ 1, and any exponent value n , we have that log a ( u n ) 5 n log a u . Naturally, we can envision situations in which the properties of logarithms might be used in conjunction within a single problem. As long as we keep in mind the precise requirements of the individual properties, we will find that this is not beyond our ability to master. EXAMPLE 11.20 Apply the properties of logarithms to fully expand the logarithmic expressions: log 5 4 5 x y       ln 7 3 2 4 x w       SOLUTION The key to successfully simplifying the first expression is to think “big picture” when looking at the argument of the logarithm. Observe that the expression is enclosed in parentheses and that it is a fraction of the form u v , where u 5 x 4 and v 5 5 y . Thus, the quotient rule of logarithms is applicable to this expression. Use of the quotient rule yields log 5 x 4 2 log 5 5 y , which we then attempt to simplify further. The first logarithm involves the variable x raised to a power, and thus the power rule of logarithms applies to this expression. The second logarithm has argument 5 y , which is a product, and thus the product rule of logarithms applies to that expression: log 5 4 5 x y       5 log 5 x 4 2 log 5 5 y by the quotient rule 5 4 log 5 x 2 (log 5 5 1 log 5 y ) by the power rule and the product rule 5 4 log 5 x 2 log 5 5 2 log 5 y by the distributive property Recall that use of the properties of logarithms is dictated by the maxim that if a rule can be used, then it must be used. When we consider the individual logarithms in this expression, we notice that the second logarithm, log 5 5, can be simplified using one of the fundamental properties of logarithms, log a a 5 1, and thus the expression simplifies once more to 5 4 log 5 x 2 1 2 log 5 y In the second logarithmic expression, the argument is, initially, an exponential expression, and thus the power rule of logarithms applies in this case.

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