Chapter 6.5, Problem 99E

To determine

To find all solutions of the equation and represent the solutions graphically.

Explanation of Solution

The given expression is

  

This can be written as

  

It means to find fifth roots of the complex number

Using the formula for nth roots of a complex number which states that,

For a positive integer the complex number has exactly distinct roots given by,

  

Where.

  

first convert the complex number into its trigonometric form.

Hence, its modulus is

  

Its argument is

  

As the complex numberlies in the fourth quadrant, so

  

Thus, the trigonometric form comes out to be

  

Using the above formula,

  

So, for roots are as follows

For 0,

  

For 1,

  

For 2,

  

For 3,