Chapter 6.5, Problem 138E

(a)

To determine

To find: The indicated root of the given expression

Answer to Problem 138E

The roots of the complex numbers are $4\left(\cos 30°+i\sin 30°\right)$ and $4\left(\cos 210°+i\sin 210°\right)$ .

Explanation of Solution

Given information:

The given expression is:

  $\sqrt[]{16}\left(\cos 60°+i\sin 60°\right)$

Formula used:

Formula of $n^{th}$ root:

  $\sqrt[n]{r}\left(\cos \frac{\theta +2\pi k}{n}+i\sin \frac{\theta +2\pi k}{n}\right)$

Consider the given expression as:

  $\sqrt[]{16}\left(\cos 60°+i\sin 60°\right)$

Consider the formula of $n^{th}$ root:

  $\sqrt[n]{r}\left(\cos \frac{\theta +2\pi k}{n}+i\sin \frac{\theta +2\pi k}{n}\right)$

Substitute $60°$ for $\theta$ in the above expression:

  $\sqrt[2]{r}\left(\cos \frac{60+2\pi k}{n}+i\sin \frac{60+2\pi k}{n}\right)$

Substitute $0$ for $k$ in the above expression:

  $\begin{array}{c}\sqrt[2]{r}\left(\cos \frac{60+2\pi \left(0\right)}{n}+i\sin \frac{60+2\pi \left(0\right)}{n}\right)\\ =4\left(\cos 30°+i\sin 30°\right)\end{array}$

Substitute $1$ for $k$ in the above expression:

  $\begin{array}{c}\sqrt[2]{r}\left(\cos \frac{60+2\pi \left(1\right)}{n}+i\sin \frac{60+2\pi \left(1\right)}{n}\right)\\ =4\left(\cos 210°+i\sin 210°\right)\end{array}$

Hence, the roots of the complex numbers are $4\left(\cos 30°+i\sin 30°\right)$ and $4\left(\cos 210°+i\sin 210°\right)$ .

(b)

To determine

To graph: The roots of the complex number from the above results.

Explanation of Solution

Given information:

The given expression is:

  $4\left(\cos 30°+i\sin 30°\right)$

  $4\left(\cos 210°+i\sin 210°\right)$

Graph:

Consider the given expression as:

  $4\left(\cos 30°+i\sin 30°\right)$

  $4\left(\cos 210°+i\sin 210°\right)$

Graph the above equations:

  

(c)

To determine

To find: The roots of the above expression in standard form

Answer to Problem 138E

The standard form of the complex numbers are $2\sqrt{3}+2i$ and $-2\sqrt{3}-2i$ .

Explanation of Solution

Given information:

The given expression is:

  $4\left(\cos 30°+i\sin 30°\right)$

  $4\left(\cos 210°+i\sin 210°\right)$

Consider the given expression as:

  $4\left(\cos 30°+i\sin 30°\right)$

  $4\left(\cos 210°+i\sin 210°\right)$

Substitute $\frac{\sqrt{3}}{2}$ for $\cos 30°$ and $\frac{1}{2}$ for $\sin 30°$ in the above expression:

  $\begin{array}{c}4\left(\cos 30°+i\sin 30°\right)=4\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)\\ =2\sqrt{3}+2i\end{array}$

Substitute $-\frac{\sqrt{3}}{2}$ for $\cos 210°$ and $-\frac{1}{2}$ for $\sin 210°$ in the above expression:

  $\begin{array}{c}4\left(\cos 210°+i\sin 210°\right)=4\left(-\frac{\sqrt{3}}{2}-\frac{1}{2}i\right)\\ =-2\sqrt{3}-2i\end{array}$

Hence, the standard form of the complex numbers are $2\sqrt{3}+2i$ and $-2\sqrt{3}-2i$ .