use DeMoivre's theorem to simplify ?

Question

How do you use DeMoivre's theorem to simplify #(-(3)^0.5+i)^(9/2)#

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Answer 1 · 2018-01-17T10:23:23

#(-sqrt3+i)^(9/2)=16-16i#

DeMoivre's theorem states that if a complex number in polar form is #a+ib=r(costheta+isintheta)#, then

#(a+ib)^n=r^n(cosntheta+isinntheta)#

Now we can write #(-(3)^0.5+i)^(9/2)# as

#(-sqrt3+i)^(9/2)#

= #{2(-sqrt3/2+i1/2)}^(9/2)#

= #{2(cos((5pi)/6)+isin((5pi)/6))}^(9/2)#

= #2^(9/2)(cos((5pi)/6xx9/2)+isin((5pi)/6xx9/2)}#

= #16sqrt2(cos((15pi)/4)+isin((15pi)/4))#

= #16sqrt2(cos(-pi/4)+isin(-pi/4))#

= #16sqrt2(1/sqrt2-i1/sqrt2)#

= #16-16i#