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Chapter 2: Problem 64

Perform the operation and write the result in standard form. $$\frac{2 i}{2+i}+\frac{5}{2-i}$$

Short Answer

Expert verified
The result in standard form is \(3 + \frac{3}{5}i\).

Step by step solution

01

Get the Complex Conjugates

First, get the complex conjugates of the denominators to simplify the fractions: the conjugate of \(2 + i\) is \(2 - i\), and the conjugate of \(2 - i\) is \(2 + i\).
02

Multiply numerators and denominators by the conjugate

Multiply the numerators and the denominators of each of the fractions by their respective complex conjugate. \[\frac{2i}{2+i} * \frac{2-i}{2-i} + \frac{5}{2-i} * \frac{2+i}{2+i}\]
03

Distribute and Simplify

Perform multiplication and simplify the fractions. \[\frac{2i(2-i)}{(2+i)(2-i)} + \frac{5(2+i)}{(2-i)(2+i)} = \frac{5 - 2i}{5} + \frac{10 + 5i}{5} \]
04

Express in Standard Form

Express the result in standard form (the a + bi form). \[1 - \frac{2}{5}i + 2 + i = 3 + \frac{3}{5}i\]

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