Chapter 6.6, Problem 79E

(a)

To determine

Thefourth roots of $16$ using formula for roots.

Answer to Problem 79E

Thefourth roots of $16$ are $z_0=2\left(\cos 0+i\sin 0\right),z_1=2\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right),z_2=2\left(\cos \pi +i\sin \pi \right)\text{ and }{z}_3=2\left(\cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2}\right)$

Explanation of Solution

Given info:

  $\sqrt[4]{625i}$

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.

Calculation:

We have

  $\begin{array}{l}a=16,b=0\\ r=\sqrt{a^2+b^2}=\sqrt{{16}^2+0^2}=16\\ \theta ={\tan }^{-1}\left(\frac{0}{16}\right)=0\\ 16=16\left(\cos 0+i\sin 0\right)\\ z_k=\sqrt[n]{r}\left(\cos \left(\frac{\theta +2\pi k}{n}\right)+i\sin \left(\frac{\theta +2\pi k}{n}\right)\right)\\ n=4,r=16,\theta =0\\ z_0=\sqrt[4]{16}\left(\cos \left(\frac{0+2\pi \times 0}{4}\right)+i\sin \left(\frac{0+2\pi \times 0}{4}\right)\right)\\ z_0=2\left(\cos 0+i\sin 0\right)\\ z_1=\sqrt[4]{16}\left(\cos \left(\frac{0+2\pi \times 1}{4}\right)+i\sin \left(\frac{0+2\pi \times 1}{4}\right)\right)\\ z_1=2\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)\\ z_2=\sqrt[4]{16}\left(\cos \left(\frac{0+2\pi \times 2}{4}\right)+i\sin \left(\frac{0+2\pi \times 2}{4}\right)\right)\\ z_2=2\left(\cos \pi +i\sin \pi \right)\\ z_3=\sqrt[4]{16}\left(\cos \left(\frac{0+2\pi \times 3}{4}\right)+i\sin \left(\frac{0+2\pi \times 3}{4}\right)\right)\\ z_3=2\left(\cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2}\right)\end{array}$

Thefourth roots of $16$ are $z_0=2\left(\cos 0+i\sin 0\right),z_1=2\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right),z_2=2\left(\cos \pi +i\sin \pi \right)\text{ and }{z}_3=2\left(\cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2}\right)$

Conclusion:

Thus,the fourth roots of $16$ are $z_0=2\left(\cos 0+i\sin 0\right),z_1=2\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right),z_2=2\left(\cos \pi +i\sin \pi \right)\text{ and }{z}_3=2\left(\cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2}\right)$

(b)

To determine

The standard form of each of the roots.

Answer to Problem 79E

The standard formof the roots are $z_0=2,z_1=2i,z_2=-2\text{ and }{z}_3=-2i$

Explanation of Solution

Given info:

Thefourth roots of $16$ are $z_0=2\left(\cos 0+i\sin 0\right),z_1=2\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right),z_2=2\left(\cos \pi +i\sin \pi \right)\text{ and }{z}_3=2\left(\cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2}\right)$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 $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$

Calculation:

The roots from part a are $z_0=2\left(\cos 0+i\sin 0\right),z_1=2\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right),z_2=2\left(\cos \pi +i\sin \pi \right)\text{ and }{z}_3=2\left(\cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2}\right)$

We have

  $\begin{array}{l}{z}_0=2\left(\cos 0+i\sin 0\right)=2\\ z_1=2\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)=2i\\ z_2=2\left(\cos \pi +i\sin \pi \right)=-2\\ z_3=2\left(\cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2}\right)=-2i\end{array}$

The standard formof the roots are

  $z_0=2,z_1=2i,z_2=-2\text{ and }{z}_3=-2i$

Conclusion:

Thus,the standard formof the roots are $z_0=2,z_1=2i,z_2=-2\text{ and }{z}_3=-2i$ .

(c)

To determine

The roots graphically.

Answer to Problem 79E

Please refer calculation part.

Explanation of Solution

Given info:

Thefourth roots of $16$ are $z_0=2,z_1=2i,z_2=-2\text{ and }{z}_3=-2i$

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 $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$

Calculation:

Graphical representation of roots are given by

  

Conclusion:

Thus,the graphical representation of roots are given.