Negative Exponents and Scientific Notation


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1 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 number in scientific notation 4. Solve an application of scientific notation NOTE By Property 2, a m a n am n when m n. Here m and n are both 5 so m n. In Section 3., we discussed exponents. We now want to extend our exponent notation to include 0 and negative integers as exponents. First, what do we do with x 0? It will help to look at a problem that gives us x 0 as a result. What if the numerator and denominator of a fraction have the same base raised to the same power and we extend our division rule? For example, a 5 () a 5 a5 5 a 0 But from our experience with fractions we know that a 5 (2) a 5 By comparing equations () and (2), it seems reasonable to make the following definition: 0 NOTE As was the case with, will be discussed in a later course. Definitions: Zero Power For any number a, a 0, a 0 In words, any expression, except 0, raised to the 0 power is. Example illustrates the use of this definition. Example Raising Expressions to the Zero Power CAUTION In part (d ) the 0 exponent applies only to the x and not to the factor 6, because the base is x. Evaluate. Assume all variables are nonzero. (a) 5 0 (b) 27 0 (c) (x 2 y) 0 if x 0 and y 0 (d) 6x if x 0 CHECK YOURSELF Evaluate. Assume all variables are nonzero. (a) 7 0 (b) ( 8) 0 (c) (xy 3 ) 0 (d) 3x 0 26
2 262 CHAPTER 3 POLYNOMIALS The second property of exponents allows us to define a negative exponent. Suppose that the exponent in the denominator is greater than the exponent in the numerator. Consider x 2 x 5 the expression. Our previous work with fractions tells us that NOTE Divide the numerator and denominator by the two common factors of x. REMEMBER: am a n am n x 2 x 5 x x x x x x x x 3 However, if we extend the second property to let n be greater than m, we have x 2 x 5 x2 5 x 3 Now, by comparing equations () and (2), it seems reasonable to define x 3 as. x In general, we have this result: 3 () (2) Definitions: Negative Powers NOTE John Wallis (66 703), an English mathematician, was the first to fully discuss the meaning of 0 and negative exponents. For any number a, a 0, and any positive integer n, a n a n Example 2 Rewriting Expressions That Contain Negative Exponents Rewrite each expression, using only positive exponents. Negative exponent in numerator (a) x 4 x 4 Positive exponent in denominator (b) m 7 m 7 (c) or 9 (d) or 000 CAUTION (e) (f) 2x 3 a 5 a 9 2 x 3 2 x 3 The 3 exponent applies only to x, because x is the base. a 5 9 a 4 a 4 (g) 4x 5 4 x 5 4 x 5
3 NEGATIVE EXPONENTS AND SCIENTIFIC NOTATION SECTION CHECK YOURSELF 2 Write, using only positive exponents. (a) a 0 (b) 4 3 (c) 3x 2 (d) x5 x 8 We will now allow negative integers as exponents in our first property for exponents. Consider Example 3. Example 3 Simplifying Expressions Containing Exponents NOTE a m a n a m n for any integers m and n. So add the exponents. Simplify (write an equivalent expression that uses only positive exponents). (a) x 5 x 2 x 5 ( 2) x 3 Note: An alternative approach would be NOTE By definition x 2 x 2 x 5 x 2 x 5 x 2 x5 x 2 x3 (b) a 7 a 5 a 7 ( 5) a 2 (c) y 5 y 9 y 5 ( 9) y 4 y 4 CHECK YOURSELF 3 Simplify (write an equivalent expression that uses only positive exponents). (a) x 7 x 2 (b) b 3 b 8 Example 4 shows that all the properties of exponents introduced in the last section can be extended to expressions with negative exponents. Example 4 Simplifying Expressions Containing Exponents Simplify each expression. (a) (b) m 3 m 4 a 2 b 6 a 5 b 4 m 3 4 m 7 m 7 a 2 5 b 6 ( 4) a 7 b 0 b0 a 7 Property 2 Apply Property 2 to each variable.
4 264 CHAPTER 3 POLYNOMIALS NOTE This could also be done by using Property 4 first, so (c) (2x 4 ) 3 (2x 4 ) 3 Definition of the negative exponent (2x 4 ) (x 4 ) x x 2 8x (x 4 ) 3 8x 2 Property 4 Property 3 (d) (y 2 ) 4 (y 3 ) 2 y 8 y 6 y 8 ( 6) Property 3 Property 2 y 2 y 2 CHECK YOURSELF 4 Simplify each expression. x 5 m 3 n 5 (a) (b) (c) (3a 3 ) 4 (d) (r3 ) 2 x 3 m 2 n 3 (r 4 ) 2 Let us now take a look at an important use of exponents, scientific notation. We begin the discussion with a calculator exercise. On most calculators, if you multiply 2.3 times 000, the display will read 2300 Multiply by 000 a second time. Now you will see Multiplying by 000 a third time will result in the display NOTE This must equal 2,300,000, or 2.3 E09 And multiplying by 000 again yields NOTE Consider the following table: , , or 2.3 E2 Can you see what is happening? This is the way calculators display very large numbers. The number on the left is always 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 frequently in science. It is not uncommon in scientific applications of algebra to find yourself working with very large or very small numbers. Even in the time of Archimedes ( B.C.E.), the study of such numbers was not unusual. Archimedes estimated that the universe was 23,000,000,000,000,000 m in diameter, which is the approximate distance light travels in 2 years. By comparison, Polaris (the North Star) is 2 actually 680 lightyears from the earth. Example 6 will discuss the idea of lightyears.
5 NEGATIVE EXPONENTS AND SCIENTIFIC NOTATION SECTION In scientific notation, Archimedes s estimate for the diameter of the universe would be m In general, we can define scientific notation as follows. Definitions: Scientific Notation Any number written in the form a 0 n in which a 0 and n is an integer, is written in scientific notation. Example 5 Using Scientific Notation Write each of the following numbers in scientific notation. NOTE Notice the pattern for writing a number in scientific notation. (a) 20, places (b) 88,000, The power is 5. 7 places The power is 7. NOTE The exponent on 0 shows the number of places we must move the decimal point. A positive exponent tells us to move right, and a negative exponent indicates to move left. (c) 520,000, places (d) 4,000,000, places (e) places If the decimal point is to be moved to the left, the exponent will be negative. NOTE To convert back to standard or decimal form, the process is simply reversed. (f) places CHECK YOURSELF 5 Write in scientific notation. (a) 22,000,000,000,000,000 (b) (c) 5,600,000 (d)
6 266 CHAPTER 3 POLYNOMIALS Example 6 An Application of Scientific Notation (a) Light travels at a speed of meters per second (m/s). There are approximately s in a year. How far does light travel in a year? We multiply the distance traveled in s by the number of seconds in a year. This yields NOTE Notice that ( )( ) ( )( ) Multiply the coefficients, and add the exponents. For our purposes we round the distance light travels in year to 0 6 m. This unit is called a lightyear, and it is used to measure astronomical distances. (b) The distance from earth to the star Spica (in Virgo) is m. How many lightyears is Spica from earth? Spica m Earth NOTE We divide the distance (in meters) by the number of meters in lightyear lightyears CHECK YOURSELF 6 The farthest object that can be seen with the unaided eye is the Andromeda galaxy. This galaxy is m from earth. What is this distance in lightyears? CHECK YOURSELF ANSWERS 3. (a) ; (b) ; (c) ; (d) 3 2. (a) ; (b) ; (c) ; (d) 4 3 or a 0 64 x 2 x 3 m 5 3. (a) x 5 ; (b) 4. (a) x 8 ; (b) ; (c) ; (d) r 2 b 5 n 8 8a 2 5. (a) ; (b) ; (c) ; (d) ,300,000 lightyears
7 Name 3.2 Exercises Section Date Evaluate (assume the variables are nonzero) ( 7) 0 3. ( 29) (x 3 y 2 ) m 0 7. x 0 8. (2a 3 b 7 ) 0 9. ( 3p 6 q 8 ) x 0 Write each of the following expressions using positive exponents; simplify when possible. ANSWERS b 8 2. p x a 2 2. (5x) 22. (3a) x x ( 2x) (3x) Use Properties and 2 to simplify each of the following expressions. Write your answers with positive exponents only. 27. a 5 a m 5 m x 8 x a 2 a 8 3. b 7 b 32. y 5 y x 0 x r 3 r a 8 a
8 ANSWERS m 9 m r r 5 x 7 x 9 x 3 x 5 a 3 a 0 x 4 x p 6 p Simplify each of the following expressions. Write your answers with positive exponents only. m 5 n 3 p 3 q (2a 3 ) 4 m 4 n 5 p 4 q (3x 2 ) (x 2 y 3 ) (a 5 b 3 ) (r 2 ) 3 (y 3 ) r 4 (m 4 ) 3 (a 3 ) 2 (a 4 ) (m 2 ) 4 (a 3 ) 3 y 6 (x 3 ) 3 (x 4 ) 2 (x 2 ) 3 (x 2 ) (x 2 ) In exercises 55 to 58, express each number in scientific notation The distance from the earth to the sun: 93,000,000 mi The diameter of a grain of sand: m. 57. The diameter of the sun: 30,000,000,000 cm. 58. The number of molecules in 22.4 L of a gas: 602,000,000,000,000,000,000,000 (Avogadro s number). 59. The mass of the sun is approximately kg. If this were written in standard or decimal form, how many 0s would follow the digit 8? 268
9 ANSWERS 60. Archimedes estimated the universe to be millimeters (mm) in diameter. If this number were written in standard or decimal form, how many 0s would follow the digit 3? In exercises 6 to 64, write each expression in standard notation In exercises 65 to 68, write each of the following in scientific notation In exercises 69 to 72, compute the expressions using scientific notation, and write your answer in that form. 69. (4 0 3 )(2 0 5 ) 70. ( )(4 0 2 ) In exercises 73 to 78, perform the indicated calculations. Write your result in scientific notation (2 0 5 )(4 0 4 ) 74. ( )(3 0 5 ) ( )(6 0 5 ) (. 0 8 )(3 0 6 ) In 975 the population of Earth was approximately 4 billion and doubling every 35 years. The formula for the population P in year Y for this doubling rate is P (in billions) 4 2 (Y 975) What was the approximate population of Earth in 960? 80. What will Earth s population be in 2025? The United States population in 990 was approximately 250 million, and the average growth rate for the past 30 years gives a doubling time of 66 years. The above formula for the United States then becomes P (in millions) (Y 990) What was the approximate population of the United States in 960? (6 0 2 )( ) ( )(3 0 2 ) 82. What will be the population of the United States in 2025 if this growth rate continues?
10 ANSWERS a. b. c. d. e. f. g. h. 83. Megrez, the nearest of the Big Dipper stars, is m from Earth. Approximately how long does it take light, traveling at 0 6 m/year, to travel from Megrez to Earth? 84. Alkaid, the most distant star in the Big Dipper, is m from Earth. Approximately how long does it take light to travel from Alkaid to Earth? 85. The number of liters (L) of water on Earth is 5,500 followed by 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 6 billion. 86. If there are people on Earth and there is enough freshwater to provide each person with L, how much freshwater is on Earth? 87. The United States uses an average of L of water per person each year. The United States has people. How many liters of water does the United States use each year? Getting Ready for Section 3.3 [Section.6] Combine like terms where possible. (a) 8m 7m (b) 9x 5x (c) 9m 2 8m (d) 8x 2 7x 2 (e) 5c 3 5c 3 (f) 9s 3 8s 3 (g) 8c 2 6c 2c 2 (h) 8r 3 7r 2 5r 3 Answers b ,000 x 5x x a x x a x 5 b 4 x m 9 6 x 4 4. x r 8 n 8 a 2 y 6 r 2 x 53. a mi cm billion million years L; L L a. 5m b. 4x c. 9m 2 8m d. x 2 e. 20c 3 f. 7s 3 g. 0c 2 6c h. 3r 3 7r 2 270
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