Rules for Exponents and the Reasons for Them


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1 Print this page Chapter 6 Rules for Exponents and the Reasons for Them 6.1 INTEGER POWERS AND THE EXPONENT RULES Repeated addition can be expressed as a product. For example, Similarly, repeated multiplication can be expressed as a power. For example, Here, 2 is called the base and 5 is called the exponent. Notice that 2 5 is not the same as 5 2, because 2 5 = = 32 but 5 2 = 5 5 = 25. In general, if a is any number and n is a positive integer, then we define Notice that a 1 = a, because here we have only 1 factor of a. For example, 5 1 = 5. We call a 2 the square of a and a 3 the cube of a. Multiplying and Dividing Powers with the Same Base When we multiply powers with the same base, we can add the exponents to get a more compact form. For example, = (5 5) (5 5 5) = = 5 5. In general, Thus, Example 1 Write with a single exponent: q5 q n 2 m 3n of 22
2 (x + y) 2 (x + y) 3. Using the rule a n a m = a n + m we have q5 q 7 = q = q = = 6 5 2n 2 m = 2 n + m 3n 3 4 = 3 n + 4 (x + y) 2 (x + y) 3 = (x + y) = (x + y) 5. Just as we applied the distributive law from left to right as well as from right to left, we can use the rule a n a m = a n + m written from right to left as a n + m = a n a m. Example 2 Write as a product: a x r + 4 y t + c (z + 2) z + 2. Using the rule a n + m = a n a m we have a = a = 25 5 a x r + 4 = x r x 4 y t + c = y t y c (z + 2) z + 2 = (z + 2) z (z + 2) 2. When we divide powers with a common base, we subtract the exponents. For example, when we divide 5 6 by 5 2, we get 2 of 22
3 More generally, if n > m, Thus, Example 3 Write with a single exponent:, where n > 4. Since we have. Just as with the products, we can write in reverse as. 3 of 22
4 Example 4 Write as a quotient: k e b  4 z w  s (p + q) a  b. Since we have Raising a Power to a Power When we take a number written in exponential form and raise it to a power, we multiply the exponents. For example, More generally, Thus, Example 5 Write with a single exponent: (q 7 ) 5 (7 p ) 3 4 of 22
5 (y a ) b (2 x ) x (f). Using the rule (a m ) n = a m n we have (q 7 ) 5 = q 7 5 = q 35 (7 p ) 3 = 7 3p (y a ) b = y ab (f). Example 6 Write as a power raised to a power: x e 4t. Using the rule a m n = (a m ) n we have = (2 3 ) 2. This could also have been written as (2 2 ) x = (4 3 ) x, which simplifies to 64 x. This could also have been written as (4 x ) 3. e 4t = (e 4 ) t. This could also have been written as (e t ) 4. Products and Quotients Raised to the Same Exponent 5 of 22
6 When we multiply we can change the order of the factors and rewrite it as = (5 5) (4 4) = = (5 4) (5 4) = (5 4) 2 = Sometimes, we want to use this process in reverse: 10 2 = (2 5) 2 = In general, Thus, Example 7 Write without parentheses: (qp) 7 (3x) n (4ab 2 ) 3 (2x 2n ) 3n. Using the rule (ab) n = a n b n we have (qp) 7 = q 7 p 7 (3x) n = 3 n x n (4ab 2 ) 3 = 4 3 a 3 (b 2 ) 3 = 64a 3 b 6. Example 8 Write with a single exponent: c 4 d 4 2n 3 n. 4x 2 a 4 (b + c) 4 (x 2 + y 2 ) 5 (c  d) 5. Using the rule a n b n = (ab) n we have c 4 d 4 = (cd) 4 6 of 22
7 2n 3 n = (2 3) n = 6 n 4x 2 = 2 2 x 2 = (2x) 2 a 4 (b + c) 4 = (a(b + c)) 4. Division of two powers with the same exponent works the same way as multiplication. For example, Or, reversing the process, More generally, Thus, Example 9 Write without parentheses:. Using the rule we have 7 of 22
8 . Example 10 Write with a single exponent: Using the rule we have. 8 of 22
9 Zero and Negative Integer Exponents We have seen that 4 5 means 4 multiplied by itself 5 times, but what is meant by 4 0, 41 or 42? We choose definitions for exponents like 0, 1, 2 that are consistent with the exponent rules. If a 0, the exponent rule for division says But, so we define a 0 = 1 if a 0. The same idea tells us how to define negative powers. If a 0, the exponent rule for division says But, so we define a 1 = 1/a. In general, we define Note that a negative exponent tells us to take the reciprocal of the base and change the sign of the exponent, not to make the number negative. Example 11 Evaluate: (2) 3 Any nonzero number to the zero power is one, so 5 0 = 1. We have We have We have 9 of 22
10 We have With these definitions, we have the exponent rule for division, where n and m are integers. Example 12 Rewrite with only positive exponents. Assume all variables are positive. We have We have We have We have 10 of 22
11 In part of Example 12, we saw that the x 2 in the denominator ended up as x 2 in the numerator. In general: Example 13 Write each of the following expressions with only positive exponents. Assume all variables are positive Summary of Exponent Rules We summarize the results of this section as follows. general 11 of 22
12 Expressions with a Common Base If m and n are integers, 1. an a m = a n + m (a m ) n = a m n Expressions with a Common Exponent If n is an integer, 1. (ab) n = a n b n 2. Zero and Negative Exponents If a is any nonzero number and n is an integer, then: a 0 = 1 Common Mistakes Be aware of the following notations that are sometimes confused: For example, 2 4 = (2 4 ) = 16, but (2) 4 = (2)(2)(2)(2) = 16. Example 14 Evaluate the following expressions for x = 2 and y = 3: (xy) 4 xy 2 (x + y) 2 x y 12 of 22
13 4x 3 (f) y 2. (2 3) 4 = (6) 4 = (6)(6)(6)(6) = (2) (3) 2 = 2 9 = 18. (2 + 3) 2 = (1) 2 = 1. (2) 3 = (2)(2)(2) = (2) 3 = 4(2)(2)(2) = 32. (f) (3) 2 = 9. Problems for Section 6.1 EXERCISES Evaluate the expressions in Exercises 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 without using a calculator (2) (5)3 (2) 2 13 of 22
14 (3) 2 (2 3 ) In Exercises 13, 14, 15, 16, 17, 18, 19, 20, 21 and 22, evaluate the following expressions for x = 2, y = 3, and z = xyz y x 15. y x / x z 19. x z of 22
15 /1000 In Exercises 23, 24, 25, 26, 27, 28, 29, 30 and 31, write the expression in the form x n, assuming x x3 x x (x4 x) 2 x x (x 3 ) x of 22
16 x 5 In Exercises 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44 and 45, write with a single exponent n n n a 5 b (a/b) x n  m 38. A n + 3 B n B B a B a (x 2 + y) 3 (x + y 2 ) B 2a + 1 (x + y) y (g + h) of 22
17 45. (a + b) 3 Without a calculator, decide whether the quantities in Exercises 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58 and 59 are positive or negative (4) 3 Negative 48. (3) Negative 50. (23) Negative (5) 2 Positive (4) 3 Negative 56. (73) Negative 58. (47) of 22
18 32p (61) 42 Positive In Exercises 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71 and 72, write each expression without parentheses. Assume all variables are positive c 12 /d r 6 /125s g 10 /49h (cf) (2p) of 22
19 t b 4t x e 4x 72. PROBLEMS In Problems 73, 74, 75, 76 and 77, decide which expressions are equivalent. Assume all variables are positive ,, equivalent;, equivalent of 22
20 x r,, equivalent;, equivalent , equivalent;,, equivalent In Problems 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90 and 91, write each expression as a product or a quotient. Assume all variables are positive of 22
21 79. a a = a 4 a 80. e 2 + r z 10 4 /10 z 82. k a  b p p a (p + q) a  b 85. (n) a + b (n) a (n) b 86. x a + b p 1  (a + b) p/(p a p b ) 88. (r  s) t + z (p + q) a /(p + q) b 91. (x + 1) ab + c 90. e t  1 (t + 1) (x + 1) ab (x + 1) c In Problems 92, 93, 94, 95, 95, 96, 97 and 98, write each expression as a power raised to a power. There may be more than one correct answer of 22
22 x (2 3 ) x = 8 x y a e 2t 98. (x + 3) 2w 99. If 3 a = w, express 3 3a in terms of w. w If 3 x = y, express 3 x + 2 in terms of y If 4 b = c, express 4 b  3 in terms of c. c/ If, and z = x c, what is a? Copyright 2010 John Wiley & Sons, Inc. All rights reserved. 22 of 22
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