Chapter 6.6, Problem 57E

To determine

The indicated power of the complex numberin standard form using DeMoivre’s Theorem.

Answer to Problem 57E

Theindicated power of the complex numberin standard form is $z=608.02+144.69i$

Explanation of Solution

Given info:

  $z={\left[5\left(\cos 3.2+i\sin 3.2\right)\right]}^4$

Formula used:

DeMoivre’s Theorem:

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

  $z^n={\left(r\left(\cos \theta +i\sin \theta \right)\right)}^n=r^n\left(\cos n\theta +i\sin n\theta \right)$

Calculation:

We have

  $\begin{array}{l}z={\left[5\left(\cos 3.2+i\sin 3.2\right)\right]}^4\\ {\left(r\left(\cos \theta +i\sin \theta \right)\right)}^n=r^n\left(\cos n\theta +i\sin n\theta \right)\\ z={\left[5\left(\cos 3.2+i\sin 3.2\right)\right]}^4=5^4\left(\cos \left(4\times 3.2\right)+i\sin \left(4\times 3.2\right)\right)\\ z=625\left(\cos 12.8+i\sin 12.8\right)\\ z=608.02+144.69i\end{array}$

The indicated power of the complex numberin standard form is $z=608.02+144.69i$

Conclusion:

Thus,theindicated power of the complex numberin standard form is $z=608.02+144.69i$