Section 1: Exponents and Roots

# Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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• davidvictor
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Section 8.1 Exponents and Roots 751 Version: Fall 2007 8.1 Exponents and Roots Before defining the next family of functions, the exponential functions , we will need to discuss exponent notation in detail. As we shall see, exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots (such as square roots and cube roots). Along the way, we’ll define higher roots and develop a few of their properties. More detailed work with roots will then be taken up in the next chapter. Integer Exponents Recall that use of a positive integer exponent is simply a shorthand for repeated mul- tiplication. For example, 5 2 = 5 · 5 (1) and 2 3 = 2 · 2 · 2 . (2) In general, b n stands for the quanitity b multiplied by itself n times. With this definition, the following Laws of Exponents hold. Laws of Exponents 1. b r b s = b r + s 2. b r b s = b r s 3. ( b r ) s = b rs The Laws of Exponents are illustrated by the following examples. l⚏ Example 3. a) 2 3 2 2 = (2 · 2 · 2)(2 · 2) = 2 · 2 · 2 · 2 · 2 = 2 5 = 2 3+2 b) 2 4 2 2 = 2 · 2 · 2 · 2 2 · 2 = 2 · 2 · 2 · 2 2 · 2 = 2 · 2 = 2 2 = 2 4 2 c) (2 3 ) 2 = (2 3 )(2 3 ) = (2 · 2 · 2)(2 · 2 · 2) = 2 · 2 · 2 · 2 · 2 · 2 = 2 6 = 2 3 · 2 Note that the second law only makes sense for r > s , since otherwise the exponent r s would be negative or 0. But actually, it turns out that we can create definitions for negative exponents and the 0 exponent, and consequently remove this restriction. Copyrighted material. See: 1

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752 Chapter 8 Exponential and Logarithmic Functions Version: Fall 2007 Negative exponents, as well as the 0 exponent, are simply defined in such a way that the Laws of Exponents will work for all integer exponents. For the 0 exponent, the first law implies that b 0 b 1 = b 0+1 , and therefore b 0 b = b . If b = 0 , we can divide both sides by b to obtain b 0 = 1 (there is one exception: 0 0 is not defined). For negative exponents, the second law implies that b n = b 0 n = b 0 b n = 1 b n , provided that b = 0 . For example, 2 3 = 1 / 2 3 = 1 / 8 , and 2 4 = 1 / 2 4 = 1 / 16 . Therefore, negative exponents and the 0 exponent are defined as follows: Definition 4. b n = 1 b n and b 0 = 1 provided that b = 0 . l⚏ Example 5. Compute the exact values of 4 3 , 6 0 , and ( 1 5 ) 2 . a) 4 3 = 1 4 3 = 1 64 b) 6 0 = 1 c) 1 5 2 = 1 ( 1 5 ) 2 = 1 1 25 = 25 We now have b n defined for all integers n , in such a way that the Laws of Exponents hold. It may be surprising to learn that we can likewise define expressions using rational exponents, such as 2 1 / 3 , in a consistent manner. Before doing so, however, we’ll need to take a detour and define roots . Roots Square Roots : Let’s begin by defining the square root of a real number. We’ve used the square root in many sections in this text, so it should be a familiar concept. Nevertheless, in this section we’ll look at square roots in more detail.
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