given that 1-i(square root(3)) is one of the cube roots of x, find x and then find its other cube roots

Question

e.mccormick · Answer

So, $1-i\sqrt{3}= \sqrt[3]{ x}$ is given.  Where would you go from there?

anonymous · Answer

would I cube both sides?

e.mccormick · Answer

I think that is a really good apprach.

anonymous · Answer

so x=1-i3sqrt{3}?

e.mccormick · Answer

hmm.... I got something I did not expect.  Going to need to check my math.

e.mccormick · Answer

Ah, canceled something wrong...

I got $1-3i\sqrt{3}+9i-i(3)^{\frac{3}{2}}$

e.mccormick · Answer

first time I got 8 and I was like... ummm.... something went wrong.  LOL.

anonymous · Answer

Ah, I know what I did wrong to get what I got previously, I forgot to foil there

anonymous · Answer

haha

e.mccormick · Answer

OK... now... here is the funny thing about this.  If that is pone of the cube roots.... what does that mean the other cube roots are?

anonymous · Answer

Wait doesnt that mean all the cube roots are the same...? or am I just looking at this wrong? O.o

e.mccormick · Answer

Yep.  That was what I instantly thought too.  So I would go with that.

anonymous · Answer

Alright! Thanks for all your help! :D

e.mccormick · Answer

Also, beacause cube roots do retain signs, I am not worried about some possiblity of a $\pm\sqrt{x}$ creeping in there.  

have fun!