125 1.3 Complex Numbers Operations on Complex Numbers Products or quotients with negative radicands are simplified by first rewriting 2-a as i 2a for a positive number a. CAUTION When working with negative radicands, use the definition ! −a =i !a before using any of the other rules for radicals. In particular, the rule 2c # 2d = 2cd is valid only when c and d are not both negative. For example, consider the following. 2 -4 # 2-9 = 2i # 3i = 6i2 = -6 Correct 2-4 # 2-9 = 21-421-92 = 236 = 6 Incorrect EXAMPLE 2 Finding Products and Quotients Involving !−a Find each product or quotient. Simplify the answers. (a) 2-7 # 2-7 (b) 2-6 # 2-10 (c) 2-20 2-2 (d) 2-48 224 SOLUTION First write all square roots in terms of i. (a) 2-7 # 2-7 = i 27 # i 27 = i2 # A 27 B 2 = -1 # 7 i2 = -1; A 2aB2 = a = -7 Multiply. (b) 2-6 # 2-10 = i 26 # i 210 = i2 # 260 = -124 # 15 = -1 # 2215 = -2215 (c) 2-20 2-2 = i 220 i 22 = B20 2 = 210 Quotient rule for radicals (d) 2-48 224 = i 248 224 = i B48 24 = i 22 Quotient rule for radicals S Now Try Exercises 29, 31, 33, and 35. EXAMPLE 3 Simplifying a Quotient Involving !−a Write -8 + 2-128 4 in standard form a + bi. SOLUTION -8 + 2-128 4 = -8 + 2-64 # 2 4 Product rule for radicals = -8 + 8i22 4 2-64 = 8i = 4 A -2 + 2i 22 B 4 Factor. = -2 + 2i 22 Lowest terms; standard form S Now Try Exercise 41. Be sure to factor before simplifying.
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