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Stitz-Zeager_College_Algebra_e-book

# Hence we solve x 1 3 cost for cost to get cost

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Unformatted text preview: w = c + di for real numbers a, b, c and d. Then zw = (a + bi)(c + di). After the usual arithmetic9 we get zw = (ac − bd) + (ad + bc)i. Therefore, |zw| = = = √ √ = a2 c2 − 2abcd + b2 d2 + a2 d2 + 2abcd + b2 c2 Expand √ a2 c2 + a2 d2 + b2 c2 + b2 d2 Rearrange terms a2 (c2 + d2 ) + b2 (c2 + d2 ) = = (ac − bd)2 + (ad + bc)2 Factor (a2 + b2 ) (c2 + d2 ) √ a2 + b2 c2 + d2 Factor Product Rule for Radicals = |z ||w| Deﬁnition of |z | and |w| Hence |zw| = |z ||w| as required. Now that the Product Rule has been established, we use it and the Principle of Mathematical Induction10 to prove the power rule. Let P (n) be the statement |z n | = |z |n . Then P (1) is true since z 1 = |z | = |z |1 . Next, assume P (k ) is true. That is, assume z k = |z |k for some k ≥ 1. Our job is to show that P (k + 1) is true, namely z k+1 = |z |k+1 . As is customary with induction proofs, we ﬁrst try to reduce the problem in such a way as to use the Induction Hypothesis. = zk z = z k+1 zk Properties of Exponents |z | Product Rule |z |k |z | = = |z |k+1 Indu...
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