SOLUTION: Find the 3 cube roots of i in polar form.

    i = (1, ) = (1,90°)  in polar form  (the modulus is 1;  the argument is  ,  or  90°).



Use deMoivre formula.  The three cubic roots of " i " are



a)  (1, ) = (1,30°);



b)  (1, ) = (1, ) = (1, ) = (1,150°);



c)  (1, ) = (1, ) = (1, ) = (1,270°).

While correct, neither the notation (r,θ) nor rei*θ is used when teaching complex numbers 
in basic trigonometry courses.  That's because such notation eliminates the trigonometric functions.

i = 0 + 1i

Graph the vector whose magnitude (modulus) is r=1, whose tail is at (0,0),
and whose tip is at (0,1), and whose argument (angle) is θ=90o. 





Since the cube root is the 1/3 power:



We raise everything to the 1/3 power.  In doing so we will use deMoivre's
theorem, where we raise the magnitude (modulus 1) to the 1/3 power (i.e.,
take its cube root 1), and multiply its argument (angle) by 1/3.



Now, since there are 3 cube roots, we take three consecutive integers for n.

Let n=0



Let n=1



Let n=2

.

Edwin