Chapter 9.8, Problem 37E

To determine

To find the roots of the given equation.

Answer to Problem 37E

  $\begin{array}{l}{x}_1=\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\\ x_2=\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\\ x_3=\cos \pi +i\sin \pi \\ x_4=\cos \frac{4\pi }{3}+i\sin \frac{4\pi }{3}\\ x_5=\cos \frac{5\pi }{3}+i\sin \frac{5\pi }{3}\end{array}$

Explanation of Solution

Given:

The given equation is $x^6-1=0$

Calculation:

Simplifying the given equation

  $\begin{array}{l}{x}^6-1=0\\ x^6=1\\ x={\left(1\right)}^{\frac{1}{6}}\end{array}$

Writing the complex number in polar form

  $\begin{array}{l}x={\left(1\right)}^{\frac{1}{6}}={\left[\cos 2\pi +i\sin 2\pi \right]}^{\frac{1}{6}}\\ x=\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\end{array}$

So, the roots are-

  $\begin{array}{l}{x}_1=\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\\ x_2=\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\\ x_3=\cos \pi +i\sin \pi \end{array}$

  $\begin{array}{l}{x}_4=\cos \frac{4\pi }{3}+i\sin \frac{4\pi }{3}\\ x_5=\cos \frac{5\pi }{3}+i\sin \frac{5\pi }{3}\end{array}$