SOLUTION: Find the cube roots of 27i
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Question 1038769: Find the cube roots of 27i
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
I'll do you one better: The following is how to find the 
th root of any complex number 
.
First convert the complex number to polar form:
and
; a\ \not=\ 0)
or
Then
^{\frac{1}{n}}\ =\ \sqrt[n]{r}\left\[\cos\frac{\varphi\,+\,2\pi{k}}{n}\ +\ i\sin\frac{\varphi\,+\,2\pi{k}}{n}\right\])
for 
So let's apply this to 
Polar Form:
and since 
, 
And 
For 
the angle is
so
^{\frac{1}{3}}\ =\ \frac{3\sqrt{3}}{2}\ +\ \frac{3}{2}i)
For 
the angle is
so
^{\frac{1}{3}}\ =\ \frac{-3\sqrt{3}}{2}\ +\ \frac{3}{2}i)
For 
the angle is
so
^{\frac{1}{3}}\ =\ -3i)
John
My calculator said it, I believe it, that settles it
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