Question #d7551

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Answer 1 · 2017-12-10T14:24:59

The answer is #=-64i#

Apply De Moivre's Theorem

#(costheta+isintheta)^n=cosntheta+isinntheta#

Let the complex number be

#z=2sqrt3-2i=2(sqrt3-i)#

#=4(sqrt3/2-1/2i)#

Transforming the complex number in the trigonometric form

Therefore,

#costheta=sqrt3/2# and #sintheta=-1/2#

The angle #theta# is in the #4th# quadrant

Therfore,

#theta=-1/6pi# #[mod 2pi]#

Therefore,

#z=4(cos(-pi/6)+isin(pi/6))#

So,

#z^3=(4(cos(-pi/6)+isin(-pi/6)))^3#

#=64(cos(-3/6pi)+isin(-3/6pi))#

#=64(cos(-1/2pi)+isin(-1/2pi))#

#=64(0-i)#

#=-64i#