Chapter 6, Problem 126CR

To determine

To find: The roots of the complex number $-1024$

Answer to Problem 126CR

The required 5 roots of the given complex number are as follows:

  $\begin{array}{l}4\left(\cos \frac{\pi }{5}+i\sin \frac{\pi }{5}\right)\\ 4\left(\cos \frac{3\pi }{5}+i\sin \frac{3\pi }{5}\right)\\ 4\left(\cos \pi +i\sin \pi \right)\\ 4\left(\cos \frac{7\pi }{5}+i\sin \frac{7\pi }{5}\right)\\ 4\left(\cos \frac{9\pi }{5}+i\sin \frac{9\pi }{5}\right)\end{array}$

Explanation of Solution

Given information: The complex number $-1024$

Formula used: $n^{th}$ roots of the complex number: $\sqrt[n]{r}\left(\cos \frac{\frac{3\pi }{2}+2\pi k}{6}+i\sin \frac{\frac{3\pi }{2}+2\pi k}{6}\right);k=0,1,2,3,\mathrm{...},n-1$

Calculation: A complex number can be rewritten as $z=r\left(\cos \theta +i\sin \theta \right)$

Hence,

  $\begin{array}{l}r=\sqrt{a^2+b^2}\\ =\sqrt{{\left(-1024\right)}^2+0^{{}^2}}\\ =1024\end{array}$

Now,

  $\begin{array}{l}\tan \theta =\frac{b}{a}\\ =\frac{0}{-1024}\\ =0\end{array}$

  $\therefore \theta =\pi$ lies in II quadrant.

Now,

  $-1024=1024\left(\cos \pi +i\sin \pi \right)$

Apply the $n^{th}$ roots formula to obtain the 5 roots of the complex number.

  $\begin{array}{l}\sqrt[5]{1024}\left(\cos \frac{\pi +2\pi \left(0\right)}{5}+i\sin \frac{\pi +2\pi \left(0\right)}{5}\right)=4\left(\cos \frac{\pi }{5}+i\sin \frac{\pi }{5}\right)\\ \sqrt[5]{1024}\left(\cos \frac{\pi +2\pi \left(1\right)}{5}+i\sin \frac{\pi +2\pi \left(1\right)}{5}\right)=4\left(\cos \frac{3\pi }{5}+i\sin \frac{3\pi }{5}\right)\\ \sqrt[5]{1024}\left(\cos \frac{\pi +2\pi \left(2\right)}{5}+i\sin \frac{\pi +2\pi \left(2\right)}{5}\right)=4\left(\cos \pi +i\sin \pi \right)\\ \sqrt[5]{1024}\left(\cos \frac{\pi +2\pi \left(3\right)}{5}+i\sin \frac{\pi +2\pi \left(3\right)}{5}\right)=4\left(\cos \frac{7\pi }{5}+i\sin \frac{7\pi }{5}\right)\\ \sqrt[5]{1024}\left(\cos \frac{\pi +2\pi \left(4\right)}{5}+i\sin \frac{\pi +2\pi \left(4\right)}{5}\right)=4\left(\cos \frac{9\pi }{5}+i\sin \frac{9\pi }{5}\right)\end{array}$

To determine

To find: The roots of the complex number $-1024$ .

Answer to Problem 126CR

The simplified roots of the given complex number are as follows:

  $\begin{array}{l}3.236+2.3512i\\ -1.236+3.8044i\\ -4\\ -1.236-\mathrm{3..8044}i\\ 3.26-2.3512i\end{array}$

Explanation of Solution

Given information: The complex number $-1024$

Formula used: $n^{th}$ roots of the complex number: $\sqrt[n]{r}\left(\cos \frac{\frac{3\pi }{2}+2\pi k}{6}+i\sin \frac{\frac{3\pi }{2}+2\pi k}{6}\right);k=0,1,2,3,\mathrm{...},n-1$

Calculation: A complex number can be rewritten as $z=r\left(\cos \theta +i\sin \theta \right)$

Hence,

  $\begin{array}{l}r=\sqrt{a^2+b^2}\\ =\sqrt{{\left(-1024\right)}^2+0^{{}^2}}\\ =1024\end{array}$

Now,

  $\begin{array}{l}\tan \theta =\frac{b}{a}\\ =\frac{0}{-1024}\\ =0\end{array}$

  $\therefore \theta =\pi$ lies in II quadrant.

Now,

  $-1024=1024\left(\cos \pi +i\sin \pi \right)$

Apply the $n^{th}$ roots formula to obtain the 5 roots of the complex number.

  $\begin{array}{l}\sqrt[5]{1024}\left(\cos \frac{\pi +2\pi \left(0\right)}{5}+i\sin \frac{\pi +2\pi \left(0\right)}{5}\right)=4\left(\cos \frac{\pi }{5}+i\sin \frac{\pi }{5}\right)\\ \sqrt[5]{1024}\left(\cos \frac{\pi +2\pi \left(1\right)}{5}+i\sin \frac{\pi +2\pi \left(1\right)}{5}\right)=4\left(\cos \frac{3\pi }{5}+i\sin \frac{3\pi }{5}\right)\\ \sqrt[5]{1024}\left(\cos \frac{\pi +2\pi \left(2\right)}{5}+i\sin \frac{\pi +2\pi \left(2\right)}{5}\right)=4\left(\cos \pi +i\sin \pi \right)\\ \sqrt[5]{1024}\left(\cos \frac{\pi +2\pi \left(3\right)}{5}+i\sin \frac{\pi +2\pi \left(3\right)}{5}\right)=4\left(\cos \frac{7\pi }{5}+i\sin \frac{7\pi }{5}\right)\\ \sqrt[5]{1024}\left(\cos \frac{\pi +2\pi \left(4\right)}{5}+i\sin \frac{\pi +2\pi \left(4\right)}{5}\right)=4\left(\cos \frac{9\pi }{5}+i\sin \frac{9\pi }{5}\right)\end{array}$

Hence simplified roots are,

  $\begin{array}{l}4\left(\cos \frac{\pi }{5}+i\sin \frac{\pi }{5}\right)=3.236+2.3512i\\ 4\left(\cos \frac{3\pi }{5}+i\sin \frac{3\pi }{5}\right)=-1.236+3.8044i\\ 4\left(\cos \pi +i\sin \pi \right)=-4\\ 4\left(\cos \frac{7\pi }{5}+i\sin \frac{7\pi }{5}\right)=-1.236-\mathrm{3..8044}i\\ 4\left(\cos \frac{9\pi }{5}+i\sin \frac{9\pi }{5}\right)=3.26-2.3512i\end{array}$

To determine

To graph: The complex number $-1024$

Explanation of Solution

Given information: The complex number $-1024$

Graph: The required graph represented below,