Exercise 5.  Find all the roots in both polar and Cartesian form for each expression.

5 (c).  .    

Solution 5 (c).

See text and/or instructor's solution manual.

Given  .  Then    where  ,  ,  and  .

The primitive fourth root of unity is  ,  
and the fourth roots of unity are  .  

The principal fourth root is  

Thus, the fourth roots of    are   which can be written as  

            

                      for  .

We are done.   

The fourth roots of    can be calculated in the ordinary way.

We are done.   

Remark.  The traditional answers for the fourth roots of    are

            for  .

We are done.   

Aside.  We can let Mathematica double check our work.

The principal value is

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell