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: 88

Answer

$I=8+3i$

Work Step by Step

$E=57+67i$ $;$ $Z=9+5i$ Ohm's Law is defined by the equation $E=IZ$. In this case, $I$ is unknown. Substitute $E$ and $Z$ into Ohm's Law equation: $57+67i=I(9+5i)$ Rearrange: $I(9+5i)=57+67i$ Solve for $I$ by taking $9+5i$ to divide the right side: $I=\dfrac{57+67i}{9+5i}=...$ To find $I$, evaluate the quotient indicated. Do so by multiplying the numerator and the denominator by the complex conjugate of the denominator: $...=\dfrac{57+67i}{9+5i}\cdot\dfrac{9-5i}{9-5i}=\dfrac{(57+67i)(9-5i)}{9^{2}-(5i)^{2}}=...$ Evaluate the operations indicated: $...=\dfrac{513-285i+603i-335i^{2}}{81-25i^{2}}=\dfrac{513+318i-335i^{2}}{81-25i^{2}}=...$ Substitute $i^{2}$ by $-1$ and simplify: $...=\dfrac{513+318i-335(-1)}{81-25(-1)}=\dfrac{513+318i+335}{81+25}=...$ $...=\dfrac{848+318i}{106}=\dfrac{848}{106}+\dfrac{318}{106}i=8+3i$
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