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

5 (e).  .  

Solution 5 (e).

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

The other values are  

This solution is complements of the authors.

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