97 CHAPTER R Test Prep Concepts Examples Properties of Real Numbers For all real numbers a, b, and c, the following hold true. Closure Properties a +b is a real number. ab is a real number. Commutative Properties a +b =b +a ab =ba Associative Properties 1 a +b2 +c =a + 1b +c2 1ab2 c =a1bc2 Identity Properties There exists a unique real number 0 such that a +0 =a and 0 +a =a. There exists a unique real number 1 such that a # 1 =a and 1 # a =a. Inverse Properties There exists a unique real number -a such that a + 1 −a2 =0 and −a +a =0. If a≠0, there exists a unique real number 1 a such that a # 1 a =1 and 1 a # a =1. Distributive Properties a1b +c2 =ab +ac a1b −c2 =ab −ac Multiplication Property of Zero 0 # a =a # 0 =0 62 # 63 = 65 p5 p2 = p3 4-3 = 1 43 1m223 = m6 ax 3b 2 = x2 32 13x24 = 34x4 R.4 Integer and Rational Exponents Rules for Exponents Let r and s be rational numbers. For all positive numbers a and b, the following hold true. Product rule ar # as =ar+s Quotient rule ar as =ar−s Negative exponent a−r = 1 ar Power rules 1ar2s =ars aa bb r = ar br 1ab2r =arbr 1 + 22 is a real number. 327 is a real number. 5 + 18 = 18 + 5 -4 # 8 = 8 # 1-42 36 + 1-324 + 5 = 6 + 1-3 + 52 17 # 6220 = 716 # 202 145 + 0 = 145 and 0 + 145 = 145 -60 # 1 = -60 and 1 # 1-602 = -60 17 + 1-172 = 0 and -17 + 17 = 0 22 # 1 22 = 1 and 1 22 # 22 = 1 315 + 82 = 3 # 5 + 3 # 8 614 - 22 = 6 # 4 - 6 # 2 0 # 4 = 4 # 0 = 0
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