Chapter 9.8, Problem 30E

To determine

To solve the given equation and then graph the roots in the complex plane.

Answer to Problem 30E

  $\begin{array}{l}{x}_1=\sqrt[8]{2}\left[\cos \left(\frac{\pi }{16}\right)+i\sin \left(\frac{\pi }{16}\right)\right]\\ x_2=\sqrt[8]{2}\left[\cos \left(\frac{9\pi }{16}\right)+i\sin \left(\frac{9\pi }{16}\right)\right]\\ x_3=\sqrt[8]{2}\left[\cos \left(\frac{17\pi }{16}\right)+i\sin \left(\frac{17\pi }{16}\right)\right]\\ x_4=\sqrt[8]{2}\left[\cos \left(\frac{25\pi }{16}\right)+i\sin \left(\frac{25\pi }{16}\right)\right]\end{array}$

Explanation of Solution

Given:

The given equation is $x^4-(1+i)=0$

Calculation:

Simplifying the given equation

  $\begin{array}{l}{x}^4-(1+i)=0\\ x^4=(1+i)\\ x={\left(1+i\right)}^{\frac{1}{4}}\end{array}$

Writing the complex number in polar form

  $x={\left[\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)\right]}^{\frac{1}{4}}$

Now find the roots as follows

  $x=\sqrt[8]{2}\left[\cos \left(\frac{\pi }{16}\right)+i\sin \left(\frac{\pi }{16}\right)\right]$

So, the roots are-

  $\begin{array}{l}{x}_1=\sqrt[8]{2}\left[\cos \left(\frac{\pi }{16}\right)+i\sin \left(\frac{\pi }{16}\right)\right]\\ x_2=\sqrt[8]{2}\left[\cos \left(\frac{9\pi }{16}\right)+i\sin \left(\frac{9\pi }{16}\right)\right]\\ x_3=\sqrt[8]{2}\left[\cos \left(\frac{17\pi }{16}\right)+i\sin \left(\frac{17\pi }{16}\right)\right]\\ x_4=\sqrt[8]{2}\left[\cos \left(\frac{25\pi }{16}\right)+i\sin \left(\frac{25\pi }{16}\right)\right]\end{array}$

The following is the graph of the solutions-