Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 1 - Equations and Inequalities - 1.3 Complex Numbers - 1.3 Exercises: 87

Answer

$Z=12+8i$

Work Step by Step

$I=10+4i$ $;$ $E=88+128i$ Ohm's Law is defined by the equation $E=IZ$. In this case, $Z$ is unknown. Substitute $I$ and $E$ into Ohm's Law equation: $88+128i=(10+4i)Z$ Rearrange: $(10+4i)Z=88+128i$ Solve for $Z$ by taking $10+4i$ to divide the right side: $Z=\dfrac{88+128i}{10+4i}=...$ To find $Z$, evaluate the quotient indicated. Do so by multiplying the numerator and the denominator by the complex conjugate of the denominator: $...=\dfrac{88+128i}{10+4i}\cdot\dfrac{10-4i}{10-4i}=\dfrac{(88+128i)(10-4i)}{10^{2}-(4i)^{2}}=...$ Evaluate the operations indicated: $...=\dfrac{880-352i+1280i-512i^{2}}{100-16i^{2}}=\dfrac{880+928i-512i^{2}}{100-16i^{2}}=...$ Substitute $i^{2}$ by $-1$ and simplify: $...=\dfrac{880+928i-512(-1)}{100-16(-1)}=\dfrac{880+928i+512}{100+16}=...$ $...=\dfrac{1392+928i}{116}=\dfrac{1392}{116}+\dfrac{928}{116}i=12+8i$
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