Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 1 - Section 1.6 - Complex Numbers - 1.6 Exercises - Page 64: 78


$z=a+bi$, $w=c+di$ Therefore, $\frac{}{zw}=\frac{}{z}\times\frac{}{w}$

Work Step by Step

$z=a+bi$, $w=c+di$, prove that $\frac{}{zw}=\frac{}{z}\times\frac{}{w}$ Evaluate $zw=(a+bi)(c+di)=ac+adi+bci+bdi^{2}$ Find the conjugate by changing the sign of the imaginary part of the complex number: $\frac{}{zw}=ac-adi-bci-bdi^{2}$ $(1)$ We have: $z=a+bi$, $w=c+di$, so: $\frac{}{z}=a-bi$ $\frac{}{w}=c-di$ So $\frac{}{z} \times\frac{}{w}=(a-bi)(c-di)=ac-adi-bci+bdi^{2}$ $(2)$ From $(1)$ and $(2)$, we have: $\frac{}{zw}=\frac{}{z}\times\frac{}{w}$
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