Chapter 6.6, Problem 92E

To determine

All solutions of the equationand represent the solutions graphically.

Answer to Problem 92E

The solutions are

  $\begin{array}{l}x=1.41+1.41i,x=-0.52+1.93i,x=-1.93+0.52i,x=-1.41-1.41i,\\ x=0.52-1.93i\text{ and }x=1.93-0.52i\end{array}$

Explanation of Solution

Given info:

  $x^6+64i=0$

Formula used:

The trigonometric form of the complex number $z=a+ib$ is

  $z=r\left(\cos \theta +i\sin \theta \right)$

Where $r=\sqrt{a^2+b^2}$ and $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$

For a positive integer n the complex number $z=r\left(\cos \theta +i\sin \theta \right)$ has exactly n distinct nth roots given by

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

where k = 0, 1, 2, . . . , ,n -1.

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

Calculation:

We have

  $\begin{array}{l}{x}^6+64i=0\\ x^6=-64i\\ x=\sqrt[6]{-64i}\\ a=0,b=-64\\ r=\sqrt{a^2+b^2}=\sqrt{0^2+{\left(-64\right)}^2}=64\end{array}$

  $\begin{array}{l}\theta ={\tan }^{-1}\left(\frac{-64}{0}\right)=\frac{3\pi }{2}\\ -64i=64\left(\cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2}\right)\end{array}$

  $\begin{array}{l}{z}_k=\sqrt[n]{r}\left(\cos \left(\frac{\theta +2\pi k}{n}\right)+i\sin \left(\frac{\theta +2\pi k}{n}\right)\right)\\ n=6,r=64,\theta =\frac{3\pi }{2}\end{array}$

  $z_0=\sqrt[6]{64}\left(\cos \left(\frac{\frac{3\pi }{2}+2\pi \times 0}{6}\right)+i\sin \left(\frac{\frac{3\pi }{2}+2\pi \times 0}{6}\right)\right)$

  $z_0=2\left(\cos \frac{3\pi }{12}+i\sin \frac{3\pi }{12}\right)=1.41+1.41i$

  $\begin{array}{l}{z}_1=\sqrt[6]{64}\left(\cos \left(\frac{\frac{3\pi }{2}+2\pi \times 1}{6}\right)+i\sin \left(\frac{\frac{3\pi }{2}+2\pi \times 1}{6}\right)\right)\\ z_1=2\left(\cos \frac{7\pi }{12}+i\sin \frac{7\pi }{12}\right)=-0.52+1.93i\end{array}$

  $z_2=\sqrt[6]{64}\left(\cos \left(\frac{\frac{3\pi }{2}+2\pi \times 2}{6}\right)+i\sin \left(\frac{\frac{3\pi }{2}+2\pi \times 2}{6}\right)\right)$

  $z_2=2\left(\cos \frac{11\pi }{12}+i\sin \frac{11\pi }{12}\right)=-1.93+0.52i$

  $z_3=\sqrt[6]{64}\left(\cos \left(\frac{\frac{3\pi }{2}+2\pi \times 3}{6}\right)+i\sin \left(\frac{\frac{3\pi }{2}+2\pi \times 3}{6}\right)\right)$

  $z_3=2\left(\cos \frac{15\pi }{12}+i\sin \frac{15\pi }{12}\right)=-1.41-1.41i$

  $z_4=\sqrt[6]{64}\left(\cos \left(\frac{\frac{3\pi }{2}+2\pi \times 4}{6}\right)+i\sin \left(\frac{\frac{3\pi }{2}+2\pi \times 4}{6}\right)\right)$

  $z_4=2\left(\cos \frac{19\pi }{12}+i\sin \frac{19\pi }{12}\right)=0.52-1.93i$

  $z_5=\sqrt[6]{64}\left(\cos \left(\frac{\frac{3\pi }{2}+2\pi \times 5}{6}\right)+i\sin \left(\frac{\frac{3\pi }{2}+2\pi \times 5}{6}\right)\right)$

  $z_5=2\left(\cos \frac{23\pi }{12}+i\sin \frac{23\pi }{12}\right)=1.93-0.52i$

The solutions are

  $\begin{array}{l}x=1.41+1.41i,x=-0.52+1.93i,x=-1.93+0.52i,x=-1.41-1.41i,\\ x=0.52-1.93i\text{ and }x=1.93-0.52i\end{array}$

Graphical representation

  

Conclusion:

Thus,the solutions are

  $\begin{array}{l}x=1.41+1.41i,x=-0.52+1.93i,x=-1.93+0.52i,x=-1.41-1.41i,\\ x=0.52-1.93i\text{ and }x=1.93-0.52i\end{array}$ .