Algebra: A Combined Approach (4th Edition)

Published by Pearson
ISBN 10: 0321726391
ISBN 13: 978-0-32172-639-1

Chapter 10 - Section 10.7 - Complex Numbers - Exercise Set: 70

Answer

$\dfrac{6+5i}{6-5i}=\dfrac{11}{61}+\dfrac{60}{61}i$

Work Step by Step

$\dfrac{6+5i}{6-5i}$ Multiply the numerator and the denominator of this expression by the complex conjugate of the denominator: $\dfrac{6+5i}{6-5i}=\dfrac{6+5i}{6-5i}\cdot\dfrac{6+5i}{6+5i}=\dfrac{(6+5i)^{2}}{6^{2}-(5i)^{2}}=...$ $...=\dfrac{6^{2}+2(6)(5i)+(5i)^{2}}{36-25i^{2}}=\dfrac{36+60i+25i^{2}}{36-25i^{2}}=...$ Substitute $i^{2}$ with $-1$ and simplify: $...=\dfrac{36+60i+25(-1)}{36-25(-1)}=\dfrac{36+60i-25}{36+25}=\dfrac{11+60i}{61}=...$ $...=\dfrac{11}{61}+\dfrac{60}{61}i$
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