128 CHAPTER 1 Equations and Inequalities This screen supports the results in Example 6. EXAMPLE 6 Dividing Complex Numbers Find each quotient. Write answers in standard form. (a) 3 + 2i 5 - i (b) 3 i SOLUTION (a) 3 + 2i 5 - i = 13 + 2i2 15 + i2 15 - i2 15 + i2 Multiply by the complex conjugate of the denominator in both the numerator and the denominator. = 15 + 3i + 10i + 2i2 25 - i2 Multiply. = 13 + 13i 26 Combine like terms; i2 = -1 = 13 26 + 13i 26 a + bi c = a c + bi c = 1 2 + 1 2 i Write in lowest terms and standard form. CHECK a 1 2 + 1 2 ib15 - i2 = 3 + 2i ✓ Quotient * Divisor = Dividend (b) 3 i = 31-i2 i1-i2 -i is the conjugate of i. = -3i -i2 Multiply. = -3i 1 -i2 = -1-12 = 1 = -3i, or 0 - 3i Standard form S Now Try Exercises 73 and 79. Powers of i can be simplified using the facts i2 = -1 and i4 = 1i222 = 1-122 = 1. Powers of i i1 = i i5 = i4 # i = 1 # i = i i2 = -1 i6 = i4 # i2 = 11-12 = -1 i3 = i2 # i = 1-12 # i = -i i7 = i4 # i3 = 1 # 1-i2 = -i i4 = i2 # i2 = 1-121-12 = 1 i8 = i4 # i4 = 1 # 1 = 1 and so on. Powers of i cycle through the same four outcomes 1i, -1, -i, and 12 because i4 has the same multiplicative property as 1. Also, any power of i with an exponent that is a multiple of 4 has value 1. As with real numbers, i0 = 1. Powers of i can be found on the TI-84 Plus calculator.
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