Chapter 6.6, Problem 32E

Solution

To find: The indicated power of the complex number.

The indicated power of the complex number is $-243i$ .

Given information:

The complex number is ${\left[3\left(\cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2}\right)\right]}^5$ .

Formula used:

By the De Moivre’s Theorem,

If $z=r\left(\cos \theta +i\sin \theta \right)$ and $n$ is a positive integer then,

  $\begin{array}{c}{z}^n={\left[r\left(\cos \theta +i\sin \theta \right)\right]}^n\\ =r^2\left(\cos n\theta +i\sin n\theta \right)\end{array}$

Calculation:

From the given complex number,

  $\begin{array}{c}r=3\\ \theta =\frac{3\pi }{2}\\ n=5\end{array}$

Find the indicated power of the complex number.

  $\begin{array}{c}{\left[3\left(\cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2}\right)\right]}^5=3^5\left[\cos 5\cdot \frac{3\pi }{2}+i\sin 5\cdot \frac{3\pi }{2}\right]\\ =243\left(\cos \frac{15\pi }{2}+i\sin \frac{15\pi }{2}\right)\end{array}$

Using coterminal angles,

  $\frac{15\pi }{2}-3\left(2\pi \right)=\frac{3\pi }{2}$

The above equation can be written as follows.

  $\begin{array}{c}{\left[3\left(\cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2}\right)\right]}^5=243\left(\cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2}\right)\\ =243\left(0-i\right)\\ =-243i\end{array}$

Therefore, the indicated power of the complex number is $-243i$ .