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

5 (a).  .    

Solution 5 (a).

See text and/or instructor's solution manual.

Given  .  Then    where  ,  ,  and  .  

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

The principal cube root of    is  

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

          

                      for  .

We are done.   

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

Remark.  The traditional answers for the cube roots of    are

        ,  ,  .

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