SOLUTION: How to find a three cube root of -64i?

Question 1074317: How to find a three cube root of -64i?
Answer by math_helper(2461) (Show Source): You can put this solution on YOUR website!

Using Euler's:
y = -64*e^((i*(pi)/2))
y^(1/3) = (-64*e^((i*(pi)/2)) ) ^(1/3)
y^(1/3) = -4* ( e^(i*(pi)/2) ) ^(1/3)
Now, before multiplying through by 1/3 on the right notice that because e^(i*pi*a) = cos(pi*a) + i*sin(pi*a) the value repeats every 2k*pi, so insert a 2k*pi into the exponent:
y^(1/3) = -4[e^(i*(pi)/2 + 2k(pi)) ]^(1/3) [ eq 1 ]
Let the roots be r0, r1, r2 (corresponding to k=0,1,2):

k=0 �>

I will leave it to you to find the other two roots (set k=1 in [eq 1] and simplify , then set k=2 and simplify )
Let me know if you have any questions.

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