84 CHAPTER R Review of Basic Concepts We cannot simply write 2x2 = x for all real numbers x. For example, what if x represents a negative number such as -5? 2 x2 = 21-522 = 225 = 5 ≠x To take care of the fact that a negative value of x can produce a positive result, we use absolute value. For any real number a, the following holds true. ! a2 = 0 a0 Examples: 21-922 = 0 -9 0 = 9 and 2132 = 0 130 = 13 We can generalize this result to any even nth root. EXAMPLE 3 Converting from Radicals to Rational Exponents Write in exponential form. Assume all variable expressions represent positive real numbers. (a) 24 x5 (b) 23y (c) 10 A 25 z B 2 (d) 523 12x427 (e) 2p2 + q SOLUTION (a) 24 x5 = x5/4 (b) 23y = 13y21/2 (c) 10 A 25 z B 2 = 10z2/5 (d) 523 12x427 = 512x427/3 = 5 # 27/3x28/3 (e) 2p2 + q = 1p2 + q21/2 S Now Try Exercises 31 and 33. Evaluating n! an If n is an even positive integer, then !n an = 0 a0 . If n is an odd positive integer, then !n an =a. EXAMPLE 4 Using Absolute Value to Find Roots Find each root. (a) 2k2 (b) 23 k3 (c) 2p4 (d) 24 p4 (e) 216m8r6 (f ) 26 1-226 (g) 212k + 322 (h) 2x2 - 4x + 4 SOLUTION (a) 2k2 = k (b) 23 k3 = k (c) 2p4 = 21p222 = 0 p2 0 = p2 (d) 24 p4 = 0 p0 (e) 216m8r6 = 0 4m4r3 0 = 4m40 r3 0 (f) 26 1-226 = 0 -2 0 = 2 (g) 212k + 322 = 0 2k + 30 (h) 2x2 - 4x + 4 = 21x - 222 = 0 x - 20 S Now Try Exercises 35, 41, and 43. LOOKING AHEAD TO CALCULUS In calculus, the “power rule” for derivatives requires converting radicals to rational exponents.
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