De Moivres?

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(2-root3i)^6

The anser is supposed to be 512i

Have tried this both by binomial expansion and using De Moivres theorem, but neither method cancels out the real parts of the expression as would be neccesary to obtain a purely imaginary solution. Any ideas guise?

I might just be screwing up the binomial expansion at some point...
This is the first line of the expansion as I have it

2^6+[6.2^5.-rt3i]+[15.2^4.(-rt3i)^2]+[20.2^3.(-rt3i)^3]+[15.2^2]+[6.2.(-rt3i)^5]+(rt3i)^6

I reduced this to:

64-192rt3i-720+480rt3i+540-108rt3i-27.

But that cancels down to 180rt3i - 143.

What am I doing wrong?

Also, when using De Moivres theorem, this is what I get.

(2-rt3i)^6

The argument of 2-rt3i is rt7

so (2-rt3)^6=(rt7)^6(cos theta + isin theta)^6

343(cos6theta+isin6theta)

Tan theta=(-rt3)/2
theta=~-40.8933

so (2-rt3)^6=343(cos6(40.8933)-isin6(40.8933))

and THIS comes down to -.143 !

Anyone?

bamp ofr sol^n

maple gives me: 64-192\,\sqrt {3}i+720\,{i}^{2}-480\,\sqrt {3}{i}^{3}+540\,{i}^{4}-108
\,\sqrt {3}{i}^{5}+27\,{i}^{6}
which simplifies to: -143+180\,i\sqrt {3}

>>4

Same as I got see.  This is a question from an excercise book which doesnt show worked solutions, it just says it should come to 512i

>>5
I'm getting the -143+180i sqrt(3) too with a different program (PARI), so there must be some typo somewhere.

According to maple, the sixth roots of 512i are

(1+√3)+(√3-1)i
(1-√3)+(-√3-1)i
(-1+√3)+(√3+1)i
(-1-√3)+(-√3+1)i
2-2i
-2+2i

Also, the absolute value of 2-√3 is √7, and the absolute value of 512i is 512.  Any power of 2-√3 will have absolute value a power of √7, which 512 is not.

OP here

Conlcusion: the book is wrong.

>>8

Yeah, I was gonna suggest that :)

I'm guessing the mistake made was the "2"- it probably should have been a 1, because...
(1-√3)i)^6 = (1/2 - √3)/2 *i)^6 * 2^6 = (sin(pi/6) - cos(pi/6)*i) * 64

=(-ie^(i*pi/6))^6 * 64= (-i)^6 * (e^(i*pi*6/6)) * 65
= (-1) * (-1) * 64
=64

...which is 8^2, where 512 is 8^3; so they're at least somewhat related numbers, but if its looking for 512i then the book isn't even close to correct.

tl;dr, The given answer is definitely false.