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


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1 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 CD with an annual rate of return of r. After the CD rolled over two times, its value was P(( r) ). Which law of eponents can be used to simplify the epression? Simplify it. P( r) GETTING MORE INVOLVED 77. Writing. When we square a product, we square each factor in the product. For eample, (b) 9b. Eplain why we cannot square a sum by simply squaring each term of the sum. 78. Writing. Eplain why we define 0 to be. Eplain why 0. In this section Negative Integral Eponents Rules for Integral Eponents Converting from Scientific Notation Converting to Scientific Notation Computations with Scientific Notation 4.7 NEGATIVE EXPONENTS AND SCIENTIFIC NOTATION We defined eponential epressions with positive integral eponents in Chapter and learned the rules for positive integral eponents in Section 4.. In this section you will first study negative eponents and then see how positive and negative integral eponents are used in scientific notation. Negative Integral Eponents If is nonzero, the reciprocal of is written as. For eample, the reciprocal of is written as. To write the reciprocal of an eponential epression in a simpler way, we use a negative eponent. So. In general we have the following definition. Negative Integral Eponents If a is a nonzero real number and n is a positive integer, then a n a n. (If n is positive, n is negative.) E X A M P L E You can evaluate epressions with negative eponents on a as shown here. Simplifying epressions with negative eponents Simplify. a) b) () c) a) b) () ( ) c) Definition of negative eponent
2 4.7 Negative Eponents and Scientific Notation (47) 4 CAUTION In simplifying, the negative sign preceding the is used after is squared and the reciprocal is found. So ( ). To evaluate a n, you can first find the nth power of a and then find the reciprocal. However, the result is the same if you first find the reciprocal of a and then find the nth power of the reciprocal. For eample, or 9 9. helpful hint Just because the eponent is negative, it doesn t mean the epression is negative. Note that () 8 while () 4. So the power and the reciprocal can be found in either order. If the eponent is, we simply find the reciprocal. For eample,, 4 4, and. Because, the reciprocal of is, and we have. These eamples illustrate the following rules. Rules for Negative Eponents If a is a nonzero real number and n is a positive integer, then a n a n, a a, a n, and n a a b n b a n. E X A M P L E Using the rules for negative eponents Simplify. a) 4 b) 0 0 c) 0 You can use a to demonstrate that the product rule for eponents holds when the eponents are negative numbers. a) b) c) Rules for Integral Eponents Negative eponents are used to make epressions involving reciprocals simpler looking and easier to write. Negative eponents have the added benefit of working in conjunction with all of the rules of eponents that you learned in Section 4.. For eample, we can use the product rule to get () and the quotient rule to get y y y y.
3 44 (48) Chapter 4 Polynomials and Eponents With negative eponents there is no need to state the quotient rule in two parts as we did in Section 4.. It can be stated simply as m a a n a mn for any integers m and n. We list the rules of eponents here for easy reference. helpful hint The definitions of the different types of eponents are a really clever mathematical invention. The fact that we have rules for performing arithmetic with those eponents makes the notation of eponents even more amazing. Rules for Integral Eponents The following rules hold for nonzero real numbers a and b and any integers m and n.. a 0 Definition of zero eponent. a m a n a mn Product rule m. a a n a mn Quotient rule 4. (a m ) n a mn Power rule. (ab) n a n b n Power of a product rule. a b n a b n n Power of a quotient rule E X A M P L E helpful hint Eample (c) could be done using the rules for negative eponents and the old quotient rule: m m m m m 4 It is always good to look at alternative methods.the more tools in your toolbo the better. The product and quotient rules for integral eponents Simplify. Write your answers without negative eponents. Assume that the variables represent nonzero real numbers. a) b b b) c) m 4y d) m y a) b b b Product rule b Simplify. b) Product rule c) m m () m m 4 Definition of negative eponent Quotient rule Simplify. m 4 Definition of negative eponent 4y d) y ( ) y y 8 In the net eample we use the power rules with negative eponents.
4 4.7 Negative Eponents and Scientific Notation (49) 4 E X A M P L E 4 You can use a to demonstrate that the power rule for eponents holds when the eponents are negative integers. helpful hint The eponent rules in this section apply to epressions that involve only multiplication and division. This is not too surprising since eponents, multiplication, and division are closely related. Recall that a a a a and a b a b. The power rules for integral eponents Simplify each epression. Write your answers with positive eponents only. Assume that all variables represent nonzero real numbers. a) (a ) b) (0 ) c) 4 y a) (a ) a Power rule a a Definition of negative eponent b) (0 ) 0 ( ) Power of a product rule 0 ()() Power rule 0 00 c) 4 (4 y ) Power of a quotient rule (y ) 4 0 y 4 Definition of negative eponent Power of a product rule and power rule 4 0 y 4 Because a b a b. 4 0 y 4 0y 4 Definition of negative eponent Simplify. Converting from Scientific Notation Many of the numbers occurring in science are either very large or very small. The speed of light is 98,9,000 feet per second. One millimeter is equal to kilometer. In scientific notation, numbers larger than 0 or smaller than are written by using positive or negative eponents. Scientific notation is based on multiplication by integral powers of 0. Multiplying a number by a positive power of 0 moves the decimal point to the right: 0(.). 0 (.) 00(.) 0 (.) 000(.) 0 Multiplying by a negative power of 0 moves the decimal point to the left: 0 (.) (.) (.) (.) (.) (.)
5 4 (440) Chapter 4 Polynomials and Eponents On a graphing you can write scientific notation by actually using the power of 0 or press EE to get the letter E, which indicates that the following number is the power of 0. Note that if the eponent is not too large, scientific notation is converted to standard notation when you press ENTER. So if n is a positive integer, multiplying by 0 n moves the decimal point n places to the right and multiplying by 0 n moves it n places to the left. A number in scientific notation is written as a product of a number between and 0 and a power of 0. The times symbol indicates multiplication. For eample, and. 0 4 are numbers in scientific notation. In scientific notation there is one digit to the left of the decimal point. To convert to standard notation, move the decimal point nine places to the right:.7 0 9,70,000,000 9 places to the right Of course, it is not necessary to put the decimal point in when writing a whole number. To convert. 0 4 to standard notation, the decimal point is moved four places to the left: places to the left In general, we use the following strategy to convert from scientific notation to standard notation. Strategy for Converting from Scientific Notation to Standard Notation. Determine the number of places to move the decimal point by eamining the eponent on the 0.. Move to the right for a positive eponent and to the left for a negative eponent. E X A M P L E Converting scientific notation to standard notation Write in standard notation. a) b) 8. 0 a) Because the eponent is positive, move the decimal point si places to the right: ,00,000 b) Because the eponent is negative, move the decimal point five places to the left Converting to Scientific Notation To convert a positive number to scientific notation, we just reverse the strategy for converting from scientific notation. Strategy for Converting to Scientific Notation. Count the number of places (n) that the decimal must be moved so that it will follow the first nonzero digit of the number.. If the original number was larger than 0, use 0 n.. If the original number was smaller than, use 0 n.
6 4.7 Negative Eponents and Scientific Notation (44) 47 Remember that the scientific notation for a number larger than 0 will have a positive power of 0 and the scientific notation for a number between 0 and will have a negative power of 0. E X A M P L E To convert to scientific notation, set the mode to scientific. In scientific mode all results are given in scientific notation. Converting numbers to scientific notation Write in scientific notation. a) 7,4,00 b) c) 0 a) Because 7,4,00 is larger than 0, the eponent on the 0 will be positive: 7,4, b) Because is smaller than, the eponent on the 0 will be negative: c) There should be only one nonzero digit to the left of the decimal point: Convert to scientific notation Product rule Computations with Scientific Notation An important feature of scientific notation is its use in computations. Numbers in scientific notation are nothing more than eponential epressions, and you have already studied operations with eponential epressions in this section. We use the same rules of eponents on numbers in scientific notation that we use on any other eponential epressions. E X A M P L E 7 With a s builtin scientific notation, some parentheses can be omitted as shown below. Writing out the powers of 0 can lead to errors. Using the rules of eponents with scientific notation Perform the indicated computations. Write the answers in scientific notation. a) ( 0 )( 0 8 ) 4 0 b) 8 0 c) ( 0 7 ) a) ( 0 )( 0 8 ) b) () Quotient rule (0.) Try these computations with your Write 0. in scientific notation. Product rule c) ( 0 7 ) (0 7 ) Power of a product rule 0 Power rule Product rule
7 48 (44) Chapter 4 Polynomials and Eponents E X A M P L E 8 Converting to scientific notation for computations Perform these computations by first converting each number into scientific notation. Give your answer in scientific notation. a) (,000,000)(0.000) b) (0,000,000) ( ) a) (,000,000)(0.000) Scientific notation 0 Product rule b) (0,000,000) ( ) ( 0 7 ) ( 0 7 ) Scientific notation Power of a product rule Product rule WARMUPS True or false? Eplain your answer True. False. False 4. True False True True True 9. ( 0 9 ) True 0. ( 0 )(4 0 4 ) False 4.7 EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What does a negative eponent mean? A negative eponent means reciprocal, as in a n a n.. What is the correct order for evaluating the operations indicated by a negative eponent? The operations can be evaluated in any order.. What is the new quotient rule for eponents? The new quotient rule is a m a n a mn for any integers m and n. 4. How do you convert a number from scientific notation to standard notation? Convert from scientific notation by multiplying by the appropriate power of 0.. How do you convert a number from standard notation to scientific notation? Convert from standard notation by counting the number of places the decimal must move so that there is one nonzero digit to the left of the decimal point.. Which numbers are not usually written in scientific notation? Numbers between and 0 are not written in scientific notation. Variables in all eercises represent positive real numbers. Evaluate each epression. See Eample () () Simplify. See Eample
8 4.7 Negative Eponents and Scientific Notation (44) 49 Simplify. Write answers without negative eponents. See Eample.. 4. y y y y (y 7 ) 0 y 7. a (a ) a 8. (b )(b ) b u u u 8 0. w w w 0 8t. 4t. 4 w t w w. 4. y y 9 7y Simplify each epression. Write answers without negative eponents. See Eample 4.. ( ) 0. (y ) 4 y 8 7. (a ) a 9 8. (b ) b 0 9. ( ) (y ) y 9 4. (4 y ) y 4. (s t 4 ) s 4 t 4 4. y 4y 44. a b 7a b a ac a 8 4. c w w 4 w 4 9 Simplify. Write answers without negative eponents ( ) 0. ( ) 7. ( ) 7 ( ) 4. (ab ) ab(ab ) a b. a b a b a b 4 a b 4. 9 y y y Write each number in standard notation. See Eample ,80,000, , , ,000,000 Write each number in scientific notation. See Eample ,98, ,70,000, Perform the computations. Write answers in scientific notation. See Eample ( 0 )( 0 ) ( 0 9 )(4 0 ) ( 0 ) ( 0 ) ( 0 4 ) ( 0 4 ) 0 8. (4 0 ) ( 0 ). 0 Perform the following computations by first converting each number into scientific notation. Write answers in scientific notation. See Eample (400)(,000,000) (40,000)(4,000,000,000) (4,00,000)(0.0000) (0.0007)(4,000,000) (00) ( ) (00) 4 (0.000) (4 000) ( 90,000) ( 0, 000) (80,000) ( )( 0.00) Perform the following computations with the aid of a. Write answers in scientific notation. Round to three decimal places. 9. (. 0 )( ) ( )(4. 0 ) (. 0 4 ) (. 0 ) ( ) ( ) (. 0 )(4. 0 ) (. 0 8 )( ) (. 0 8 )( ) ( ) ( 0 99 ). 0 00
9 0 (444) Chapter 4 Polynomials and Eponents Solve each problem. 99. Distance to the sun. The distance from the earth to the sun is 9 million miles. Epress this distance in feet. ( mile 80 feet.) feet Sun 9 million miles Earth FIGURE FOR EXERCISE Speed of light. The speed of light is feet per second. How long does it take light to travel from the sun to the earth? See Eercise minutes 0. Warp drive, Scotty. How long does it take a spacecraft traveling at 0 miles per hour (warp factor 4) to travel 9 million miles hours 0. Area of a dot. If the radius of a very small circle is. 0 8 centimeters, then what is the circle s area?.7 0 cm 0. Circumference of a circle. If the circumference of a circle is feet, then what is its radius? feet 04. Diameter of a circle. If the diameter of a circle is. 0 meters, then what is its radius?. 0 meters 0. Etracting metals from ore. Thomas Sherwood studied the relationship between the concentration of a metal in commercial ore and the price of the metal. The accompanying graph shows the Sherwood plot with the locations of several metals marked. Even though the scales on this graph are not typical, the graph can be read in the same manner as other graphs. Note also that a concentration of 00 is 00%. a) Use the figure to estimate the price of copper (Cu) and its concentration in commercial ore. $ per pound and % b) Use the figure to estimate the price of a metal that has a concentration of 0 percent in commercial ore. $,000,000 per pound. c) Would the four points shown in the graph lie along a straight line if they were plotted in our usual coordinate system? No 0. Recycling metals. The accompanying graph shows the prices of various metals that are being recycled and the minimum concentration in waste required for recycling. The straight line is the line from the figure for Eercise 0. Points above the line correspond to metals for which it is economically feasible to increase recycling efforts. a) Use the figure to estimate the price of mercury (Hg) and the minimum concentration in waste required for recycling mercury. $ per pound and % b) Use the figure to estimate the price of silver (Ag) and the minimum concentration in waste required for recycling silver. $00 per pound and 0.% Metal price (dollars/pound) Pb Hg Ag Cr Ba Concentration in waste (percent) V FIGURE FOR EXERCISE Present value. The present value P that will amount to A dollars in n years with interest compounded annually at annual interest rate r, is given by P A ( r ) n. Find the present value that will amount to $0,000 in 0 years at 8% compounded annually. $0, Investing in stocks. U.S. small company stocks have returned an average of 4.9% annually for the last 0 years (T. Rowe Price, Use the present value formula from the previous eercise to find the amount invested today in small company stocks that would be worth $ million in 0 years, assuming that small company stocks continue to return 4.9% annually for the net 0 years. $9.8 Metal price (dollars/pound) Ra 0 4 Au 0 0 Cu U Concentration in ore (percent) Value (millions of dollars) 0. 0 Amount after 0 years, $ million Present value, P Years FIGURE FOR EXERCISE 0 FIGURE FOR EXERCISE 08
10 Chapter 4 Collaborative Activities (44) GETTING MORE INVOLVED 09. Eploration. a) If w 0, then what can you say about w? b) If () m 0, then what can you say about m? c) What restriction must be placed on w and m so that w m 0? a) w 0 b) m is odd c) w 0 and m odd 0. Discussion. Which of the following epressions is not equal to? Eplain your answer. a) b) c) ( ) d) () e) () e COLLABORATIVE ACTIVITIES Area as a Model of FOIL Sometimes we can use drawings to represent mathematical operations. The area of a rectangle can represent the process we use when multiplying binomials. The rectangle below represents the multiplication of the binomials ( ) and ( ): The areas of the inner rectangles are,,, and. The area of the red rectangle equals the sum of the areas of the four inner rectangles. Area of red rectangle: ( )( ) 8. a. With your partner, find the areas of the inner rectangles to find the product ( )( 7) below: ( )( 7)? b. Find the same product ( )( 7) using FOIL. For problem, student A uses FOIL to find the given product while student B finds the area with the diagram.. ( + ) ( + 8) ( + ) 8 ( + ) 7 ( + 7) ( 8)( )? Grouping: Pairs Topic: Multiplying polynomials. For problem, student B uses FOIL and A uses the diagram. ( )( 4)? 4. StudentAdraws a diagram to find the product ( )( 7). Student B finds ( )( 7) using FOIL.. Student B draws a diagram to find the product ( )( ). Student A finds ( )( ) using FOIL. Thinking in reverse: Work together to complete the product that is represented by the given diagram.. 7. ( + ) ( +?) ( +?) 4 ( +?) ( + 4) 9 ( +?) 0 ( + ) Etension: Make up a FOIL problem, then have your partner draw a diagram of it.
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