Exercise 9.  Find the three solutions to  .  

Solution 9.

See text and/or instructor's solution manual.

Given  ,  we can square both sides to obtain the equation  .  

Use  .  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 solutions to    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