Chapter 8.3, Problem 83E

To determine

To calculate: To find the indicated roots and graph the roots in the complex plane

Answer to Problem 83E

The forth roots are ${\omega }_0=3\left[\cos \frac{3\pi }{8}+i\sin \frac{3\pi }{8}\right]$ , ${\omega }_1=3\left[\cos \frac{7\pi }{8}+i\sin \frac{7\pi }{8}\right]$ , ${\omega }_2=3\left[\cos \frac{11\pi }{8}+i\sin \frac{11\pi }{8}\right]$ and ${\omega }_3=3\left[\cos \frac{15\pi }{8}+i\sin \frac{15\pi }{8}\right]$

Explanation of Solution

Given information: The forth roots of $-81i$

Formula Used:

Complex number is a number that can be expressed in the form of $a+bi$ , where $a$ and $b$ are real numbers

  $z=a+bi$

If $z=r\left(\cos \theta +isin\theta \right)$ and $n$ is a positive integer, then $z$ has the $n$ distinct $n^{th}$ roots

  ${\omega }_k=r^{\frac{1}{n}}\left[\cos \left(\frac{\theta +2\pi k}{n}\right)+i\sin \left(\frac{\theta +2\pi k}{n}\right)\right]$ for $k=0,1,2,\mathrm{3....},n-1$

Calculation:

Complex number is given as

  $z=-81i$

Polar form of complex number is

  $z=r\left(\cos \theta +isin\theta \right)--(1)$

Calculating the value of $r$ :

  $\begin{array}{l}r=\sqrt{{\left(0\right)}^2+{\left(81\right)}^2}\\ r=\sqrt{{\left(81\right)}^2}\\ r=81\end{array}$

Calculating the value of $\theta :$

  $\begin{array}{l}\tan \theta =\frac{-81}{0}\\ \theta ={\tan }^{-1}\frac{-81}{0}\\ \theta =2\pi -\frac{\pi }{2}\\ \theta =\frac{3\pi }{2}\end{array}$

Hence, complex number in polar form is

  $z=81\left(\cos \frac{3\pi }{2}+isin\frac{3\pi }{2}\right)$

In order to find fourth roots of above complex number, apply nth Rule of Complex numbers

  ${\omega }_k=r^{\frac{1}{n}}\left[\cos \left(\frac{\theta +2\pi k}{n}\right)+i\sin \left(\frac{\theta +2\pi k}{n}\right)\right]$

Here, $n=4$ and $z=81\left(\cos \frac{3\pi }{2}+isin\frac{3\pi }{2}\right)$

Thus,

  $\begin{array}{l}{\omega }_k={81}^{\frac{1}{4}}\left[\cos \left(\frac{\raisebox{1ex}{$3\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.+2\pi k}{4}\right)+i\sin \left(\frac{\raisebox{1ex}{$3\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.+2\pi k}{4}\right)\right]\\ {\omega }_k={\left(3^4\right)}^{\frac{1}{4}}\left[\cos \left(\frac{3\pi }{8}+\frac{2\pi k}{4}\right)+i\sin \left(\frac{3\pi }{8}+\frac{2\pi k}{4}\right)\right]\\ {\omega }_k=3\left[\cos \left(\frac{3\pi }{8}+\frac{\pi k}{2}\right)+i\sin \left(\frac{3\pi }{8}+\frac{\pi k}{2}\right)\right]\end{array}$

Now, the value of $k$ is from $0$ to $n-1$

Thus,

  $\begin{array}{l}0\le k\le 4-1\\ 0\le k\le 3\end{array}$

Hence, $k=0,1,2,3$

When $k=0$

  $\begin{array}{l}{\omega }_0=3\left[\cos \left(\frac{3\pi }{8}+\frac{\pi \left(0\right)}{2}\right)+i\sin \left(\frac{3\pi }{8}+\frac{\pi \left(0\right)}{2}\right)\right]\\ {\omega }_0=3\left[\cos \left(\frac{3\pi }{8}+0\right)+i\sin \left(\frac{3\pi }{8}+0\right)\right]\\ {\omega }_0=3\left[\cos \frac{3\pi }{8}+i\sin \frac{3\pi }{8}\right]\end{array}$

When $k=1$

  $\begin{array}{l}{\omega }_1=3\left[\cos \left(\frac{3\pi }{8}+\frac{\pi \left(1\right)}{2}\right)+i\sin \left(\frac{3\pi }{8}+\frac{\pi \left(1\right)}{2}\right)\right]\\ {\omega }_1=3\left[\cos \left(\frac{3\pi }{8}+\frac{\pi }{2}\right)+i\sin \left(\frac{3\pi }{8}+\frac{\pi }{2}\right)\right]\\ {\omega }_1=3\left[\cos \frac{7\pi }{8}+i\sin \frac{7\pi }{8}\right]\end{array}$

When $k=2$

  $\begin{array}{l}{\omega }_2=3\left[\cos \left(\frac{3\pi }{8}+\frac{\pi \left(2\right)}{2}\right)+i\sin \left(\frac{3\pi }{8}+\frac{\pi \left(2\right)}{2}\right)\right]\\ {\omega }_2=3\left[\cos \left(\frac{3\pi }{8}+\pi \right)+i\sin \left(\frac{3\pi }{8}+\pi \right)\right]\\ {\omega }_2=3\left[\cos \frac{11\pi }{8}+i\sin \frac{11\pi }{8}\right]\end{array}$

When $k=3$

  $\begin{array}{l}{\omega }_3=3\left[\cos \left(\frac{3\pi }{8}+\frac{\pi \left(3\right)}{2}\right)+i\sin \left(\frac{3\pi }{8}+\frac{\pi \left(3\right)}{2}\right)\right]\\ {\omega }_3=3\left[\cos \left(\frac{3\pi }{8}+\frac{3\pi }{2}\right)+i\sin \left(\frac{3\pi }{8}+\frac{3\pi }{2}\right)\right]\\ {\omega }_3=3\left[\cos \frac{15\pi }{8}+i\sin \frac{15\pi }{8}\right]\end{array}$

Now, plotting the above points on the complex plane. All points lie on a circle of radius $r=3$

  

Conclusion:

Hence, forth roots are ${\omega }_0=3\left[\cos \frac{3\pi }{8}+i\sin \frac{3\pi }{8}\right]$ , ${\omega }_1=3\left[\cos \frac{7\pi }{8}+i\sin \frac{7\pi }{8}\right]$ , ${\omega }_2=3\left[\cos \frac{11\pi }{8}+i\sin \frac{11\pi }{8}\right]$ and ${\omega }_3=3\left[\cos \frac{15\pi }{8}+i\sin \frac{15\pi }{8}\right]$