Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 2 - Section 2.1 - Complex Numbers - Exercise Set - Page 314: 27


The standard form of the expression $\frac{2+3i}{2+i}$ is $\frac{7}{5}+\frac{4}{5}i$.

Work Step by Step

Consider the expression,$\frac{2+3i}{2+i}$ Multiply by the complex conjugate of the denominator in the numerator and the denominator. $\frac{2+3i}{2+i}=\frac{\left( 2+3i \right)}{\left( 2+i \right)}\cdot \frac{\left( 2-i \right)}{\left( 2-i \right)}$ Use the FOIL method. \[\begin{align} & \frac{2+3i}{2+i}=\frac{\left( 2+3i \right)\left( 2-i \right)}{\left( 2+i \right)\left( 2-i \right)} \\ & =\frac{4-2i+6i-3{{i}^{2}}}{4-2i+2i-{{i}^{2}}} \\ & =\frac{4+4i-3{{i}^{2}}}{4-{{i}^{2}}} \end{align}\] Replace the value ${{i}^{2}}=-1$. \[\begin{align} & \frac{2+3i}{2+i}=\frac{4+4i-3\left( -1 \right)}{4-\left( -1 \right)} \\ & =\frac{4+4i+3}{4+1} \end{align}\] Make a group of real and imaginary terms. \[\begin{align} & \frac{2+3i}{2+i}=\frac{\left( 4+3 \right)+4i}{5} \\ & =\frac{7+4i}{5} \end{align}\] Express the complex number in the standard form. \[\frac{2+3i}{2+i}=\frac{7}{5}+\frac{4}{5}i\] Therefore, the standard form of the expression $\frac{2+3i}{2+i}$ is $\frac{7}{5}+\frac{4}{5}i$.
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