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360 CHAPTER 7 The Basic Concepts of Algebra 64. Target Heart Rate To achieve the maximum benefit from exercising, the heart rate in beats per minute should be in the target heart rate zone (THR). For a person aged A, the formula is .7220 A THR .85220 A. Find the THR to the nearest whole number for each age. (Source: Hockey, Robert V., Physical Fitness: The Pathway to Healthful Living, Times Mirror/ Mosby College Publishing, 1989.) (a) 35 (b) Your age 7.5 Profit/Cost Analysis A product will produce a profit only when the revenue R from selling the product exceeds the cost C of producing it. In Exercises 65 and 66 find the smallest whole number of units x that must be sold for the business to show a profit for the item described. 65. Peripheral Visions, Inc. finds that the cost to produce x studio quality videotapes is C 20x 100 , while the revenue produced from them is R 24x (C and R in dollars). 66. Speedy Delivery finds that the cost to make x deliveries is C 3x 2300, while the revenue produced from them is R 5.50x (C and R in dollars). Properties of Exponents and Scientific Notation Exponents Exponents are used to write products of repeated factors. For example, the product 3 3 3 3 is written 3 3 3 3 34.k Exponent 4 factors of 3 0. 0 .. 34 3 3 3 3 81 . The term googol, meaning 10100, was coined by Professor Edward Kasner of Columbia University. A googol is made up of a 1 with one hundred zeros following it. This number exceeds the estimated number of electrons in the universe, which is 1079. The Web search engine Google is named after a googol. Sergey Brin, president and cofounder of Google, Inc., was a math major. He chose the name Google to describe the vast reach of this search engine. (Source: The Gazette, March 2, 2001.) If a googol isn’t big enough for you, try a googolplex: googolplex 10 The number 4 shows that 3 appears as a factor four times. The number 4 is the exponent and 3 is the base. The quantity 34 is called an exponential expression. Read 34 as “3 to the fourth power,” or “3 to the fourth.” Multiplying out the four 3s gives googol . Exponential Expression If a is a real number and n is a natural number, then the exponential expression an is defined as an a a a … a . 10 1 0 000 a Base n factors of a The number a is the base and n is the exponent. EXAMPLE 1 (a) 7 7 7 49 2 Evaluate each exponential expression. Read 72 as “7 squared.” (b) 53 5 5 5 125 Read 53 as “5 cubed.” (c) 24 2222 16 (d) 25 22222 32 (e) 51 5 An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 7.5 Properties of Exponents and Scientific Notation 361 In the exponential expression 3z 7, the base of the exponent 7 is z, not 3z. That is, Base is z. 3z7 3 z z z z z z z 7 while Base is 3z. 3z 3z3z3z3z3z3z3z . To evaluate 26, the parentheses around 2 indicate that the base is 2, so 26 222222 64 . Base is 2. In the expression 2 , the base is 2, not 2. The sign tells us to find the negative, or additive inverse, of 26. It acts as a symbol for the factor 1. 6 This screen supports the results preceding Example 2. 26 2 2 2 2 2 2 64 Base is 2. Therefore, since 64 64, 2 2 . 6 EXAMPLE 2 6 Evaluate each exponential expression. (a) 42 4 4 16 (b) 84 8 8 8 8 4096 (c) 24 2 2 2 2 16 There are several useful rules that simplify work with exponents. For example, the product 25 23 can be simplified as follows. ~b k~~~~ 5 3 8 ~~~~ ~~~~~b b~~ 25 23 2 2 2 2 22 2 2 28 This result—products of exponential expressions with the same base are found by adding exponents —is generalized as the product rule for exponents. Product Rule for Exponents If m and n are natural numbers and a is any real number, then am an amn. EXAMPLE 3 Apply the product rule for exponents in each case. (a) 34 37 347 311 (b) 53 5 53 51 531 54 (c) y 3 y 8 y 2 y 382 y 13 (d) 5y 23y 4 53y 2y 4 15y 24 15y 6 Associative and commutative properties (e) 7p3q2p5q2 72p3p5qq 2 14p8q 3 So far we have discussed only positive exponents. How might we define a 0 exponent? Suppose we multiply 42 by 40. By the product rule, 42 40 420 42. An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 362 CHAPTER 7 The Basic Concepts of Algebra For the product rule to hold true, 40 must equal 1, and so we define a0 this way for any nonzero real number a. Zero Exponent If a is any nonzero real number, then a0 1. The expression 00 is undefined.* EXAMPLE 4 Evaluate each expression. (a) 120 1 (c) 60 60 1 (e) 8k0 1, k 0 This screen supports the results in parts (b), (c), and (d) of Example 4. Base is 6. (b) 60 1 Base is 6. 0 0 (d) 5 12 1 1 2 How should we define a negative exponent? Using the product rule again, 82 82 822 80 1. 1 is the reciprocal of 82, and a 82 number can have only one reciprocal. Therefore, it is reasonable to conclude that 1 82 2 . We can generalize and make the following definition. 8 This indicates that 82 is the reciprocal of 82. But Negative Exponent For any natural number n and any nonzero real number a, an 1 . an With this definition, and the ones given earlier for positive and zero exponents, the expression an is meaningful for any integer exponent n and any nonzero real number a. EXAMPLE 5 Write the following expressions with only positive exponents. (a) 23 1 1 23 8 (b) 32 1 1 32 9 (c) 61 1 1 1 6 6 (d) 5z3 1 , 5z3 z0 *In advanced treatments, 00 is called an indeterminate form. An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 7.5 (e) 5z3 5 1 5 3 3, z z (g) m2 EXAMPLE Properties of Exponents and Scientific Notation (f) 5z 23 z0 1 , 5z 23 z0 1 , m2 m0 6 Evaluate each of the following expressions. (h) m4 (a) 31 41 1 1 4 3 7 3 4 12 12 12 (b) 51 21 1 1 2 5 3 5 2 10 10 10 31 363 1 , m0 m4 1 1 and 41 3 4 1 1 1 23 1 1 23 8 23 1 23 1 23 1 1 32 32 9 23 23 3 3 (d) 2 3 1 2 1 2 8 32 (c) This screen supports the results in parts (a), (b), and (d) of Example 6. Parts (c) and (d) of Example 6 suggest the following generalizations. Special Rules for Negative Exponents If a 0 and b 0, then 1 an an and an bm . bm an A quotient, such as a8a3, can be simplified in much the same way as a product. (In all quotients of this type, assume that the denominator is not 0.) Using the definition of an exponent, a8 a a a a a a a a a a a a a a 5. a3 aaa Notice that 8 3 5. In the same way, a3 aaa 1 5 a5. 8 a aaaaaaaa a Here, 3 8 5. These examples suggest the following quotient rule for exponents. Quotient Rule for Exponents If a is any nonzero real number and m and n are nonzero integers, then am amn. an An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 364 CHAPTER 7 The Basic Concepts of Algebra EXAMPLE 7 Apply the quotient rule for exponents in each case. Numerator exponent b~ Denominator exponent b (a) 37 372 35 a 32 Minus sign 6 (b) p p62 p4, p2 (d) 74 1 46 72 2 6 7 7 7 p0 (c) 1210 12109 121 12 129 (e) k7 1 712 k 5 5, 12 k k k k0 EXAMPLE This screen supports the results in parts (a) and (d) of Example 8. 8 Write each quotient using only positive exponents. (a) 27 273 210 23 (b) 82 1 825 87 7 85 8 (c) 65 1 652 63 3 62 6 (d) 4 41 11 42 1 1 4 4 4 (e) z 5 z 58 z 3, z 8 z0 The expression 342 can be simplified as 342 34 34 344 38, where 4 2 8. This example suggests the first of the power rules for exponents; the other two parts can be demonstrated with similar examples. Power Rules for Exponents If a and b are real numbers, and m and n are integers, then amn amn, abm ambm, and a b m am b 0. bm In the statements of rules for exponents, we always assume that zero never appears to a negative power or to the power zero. EXAMPLE 9 Use a power rule in each case. 2 4 24 16 4 3 3 81 (c) 3y4 34y4 81y4 (d) 6p72 62p72 62p14 36p14 2m5 3 23m5 3 23m15 8m15 (e) , z0 z z3 z3 z3 (a) p83 p83 p24 (b) An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 7.5 Properties of Exponents and Scientific Notation Notice that 63 1 6 3 1 and 216 2 2 3 2 3 2 365 9 . 4 These are examples of two special rules for negative exponents that apply when working with fractions. Special Rules for Negative Exponents If a 0 and b 0, then an E X A M P L E 10 and then evaluate. (a) 2 7 3 1 a n a b and n b a n . Write the following expressions with only positive exponents 3 7 2 49 9 (b) 4 5 3 3 5 4 125 64 The definitions and rules of this section are summarized here. This screen supports the results of Example 10. Definitions and Rules for Exponents For all integers m and n and all real numbers a and b, the following rules apply. Product Rule am an amn Quotient Rule am amn an Zero Exponent a0 1 (a 0) Negative Exponent an Power Rules Special Rules for Negative Exponents (a 0) 1 (a 0) an amn amn am a m m b b abm ambm (b 0) 1 an bm n (a 0) n a m a b an 1 n an (a 0) a a n b n (a, b 0) b a An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. (a, b 0) 366 CHAPTER 7 The Basic Concepts of Algebra E X A M P L E 11 Simplify each expression so that no negative exponents appear in the final result. Assume all variables represent nonzero real numbers. (a) 32 35 325 33 1 33 1 27 or (b) x 3 x 4 x 2 x 342 x 5 (c) 425 425 410 (e) x 4y 2 x 4 y 2 2 5 x 2y 5 x y 1 x5 (d) x 46 x 46 x 24 (f ) 23x 22 232 x 22 x 42 y 25 26x 4 x 6y 7 1 x 24 y7 x6 x4 26 or x4 64 Scientific Notation Many of the numbers that occur in science are very large, such as the number of one-celled organisms that will sustain a whale for a few hours: 400,000,000,000,000. Other numbers are very small, such as the shortest wavelength of visible light, about .0000004 meter. Writing these numbers is simplified by using scientific notation. Scientific Notation A number is written in scientific notation when it is expressed in the form a 10n, where 1 a 10, and n is an integer. As stated in the definition, scientific notation requires that the number be written as a product of a number between 1 and 10 (or 1 and 10) and some integer power of 10. (1 and 1 are allowed as values of a, but 10 and 10 are not.) For example, since 8000 8 1000 8 10 3, the number 8000 is written in scientific notation as 8000 8 103. When using scientific notation, it is customary to use instead of a dot to show multiplication. The steps involved in writing a number in scientific notation follow. (If the number is negative, ignore the minus sign, go through these steps, and then attach a minus sign to the result.) An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 7.5 Properties of Exponents and Scientific Notation 367 Converting to Scientific Notation Step 1: Position the decimal point. Place a caret, ^, to the right of the first nonzero digit, where the decimal point will be placed. Step 2: Determine the numeral for the exponent. Count the number of digits from the decimal point to the caret. This number gives the absolute value of the exponent on 10. Step 3: Determine the sign for the exponent. Decide whether multiplying by 10 n should make the result of Step 1 larger or smaller. The exponent should be positive to make the result larger; it should be negative to make the result smaller. It is helpful to remember that for n 1, 10n 1 and 10n 10. EXAMPLE notation. 12 Convert each number from standard notation to scientific (a) 820,000 Place a caret to the right of the 8 (the first nonzero digit) to mark the new location of the decimal point. If a graphing calculator is set in scientific notation mode, it will give results as shown here. E5 means “times 105” and E5 means “times 105”. Compare to the results of Example 12. 8 20,000 ^ Count from the decimal point, which is understood to be after the last 0, to the caret. 8 20,000. k Decimal point ^ Count 5 places. Since the number 8.2 is to be made larger, the exponent on 10 is positive. 820,000 8.2 10 5 (b) .000072 Count from left to right. .00007 2 ^ 5 places Since the number 7.2 is to be made smaller, the exponent on 10 is negative. .000072 7.2 105 To convert a number written in scientific notation to standard notation, just work in reverse. An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 368 CHAPTER 7 The Basic Concepts of Algebra Converting from Scientific Notation to Standard Notation Multiplying a number by a positive power of 10 makes the number larger, so move the decimal point to the right if n is positive in 10 n. Multiplying by a negative power of 10 makes a number smaller, so move the decimal point to the left if n is negative. If n is zero, leave the decimal point where it is. EXAMPLE 13 Write each number in standard notation. (a) 6.93 105 6.93000 5 places The decimal point was moved 5 places to the right. (It was necessary to attach 3 zeros.) 6.93 10 5 693,000 (b) 4.7 106 000004.7 Attach 0s as necessary. 6 places The decimal point was moved 6 places to the left. Therefore, 4.7 10 6 .0000047. (c) 1.083 100 1.083 We can use scientific notation and the rules for exponents to simplify calculations. 1,920,000 .0015 by using scientific notation. .000032 45,000 First, express all numbers in scientific notation. EXAMPLE 14 Evaluate 1,920,000 .0015 1.92 106 1.5 103 .000032 45,000 3.2 105 4.5 104 1.92 1.5 106 103 3.2 4.5 105 104 1.92 1.5 104 3.2 4.5 .2 104 2 101 104 Commutative and associative properties Product and quotient rules Simplify. 2 103 2000 An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 7.5 Properties of Exponents and Scientific Notation 369 E X A M P L E 15 In 1990, the national health care expenditure was $695.6 billion. By 2000, this figure had risen by a factor of 1.9; that is, it almost doubled in only 10 years. (Source: U.S. Centers for Medicare & Medicaid Services.) (a) Write the 1990 health care expenditure using scientific notation. 695.6 billion 695.6 109 6.956 102 109 6.956 1011 Product rule In 1990, the expenditure was $6.956 1011. (b) What was the expenditure in 2000? Multiply the result in part (a) by 1.9. 6.956 1011 1.9 1.9 6.956 1011 13.216 1011 Commutative and associative properties Round to three decimal places. The 2000 expenditure was $1,321,600,000,000, over $1 trillion. An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 25 25 9 9 25 9 25 54 10 3 25 54 23 32 34 52 7 2 41 7 2 41 3 2 43 2 32 3 2 3 1 5 1 5 42 2 2 5 1 2 3