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 quotient-power 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. Quotient-Power 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 light-ears from the earth. Eample 0 will discuss the idea of light-ears.
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 light-ear. ( )(. 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 light-ear, 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 light-ears is Spica from earth? light-ears 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 light-ears? 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 light-ears.
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 8-da 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?