Chapter 12.7, Problem 33E

To determine

To find: the exponential form of each complex number.

Answer to Problem 33E

  $2e^{i\frac{3\pi }{4}}$

Explanation of Solution

Given information:

  $-\sqrt{2}+\sqrt{2}i$

Calculation:

First evaluate $r$ and then $\theta$ as shown below.

  $\begin{array}{c}r=\sqrt{{\left(-\sqrt{2}\right)}^2+{\left(\sqrt{2}\right)}^2}\left[\text{Using }r=\sqrt{a^2+b^2}\right]\\ =2\\ \theta =\arctan \left(\frac{\sqrt{2}}{-\sqrt{2}}\right)\left[\text{Using }\theta =\arctan \frac{b}{a}\right]\\ =\frac{3\pi }{4}\end{array}$

Use $r{e}^{i\theta }=-\sqrt{2}+\sqrt{2}i$ to convert the complex number into exponential form as follows:

  $\begin{array}{c}r{e}^{i\theta }=-\sqrt{2}+\sqrt{2}i\\ =2e^{i\frac{3\pi }{4}}\end{array}$

Thus, the value is $2e^{i\frac{3\pi }{4}}$ .