How do you use DeMoivre's Theorem to simplify #(2+i)^5#?

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Answer 1 · 2016-12-03T15:11:00

The answer is #=55.9(-0.68+0.73i)#

To use DeMoivre's theorem, we need to change to the trigonometric form of the complex numbers.

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

Here,

#z=2+i#

#∥z∥=sqrt(4+1)=sqrt5#

#z=sqrt5(2/sqrt5+i/sqrt5)#

#z=sqrt5(costheta+isintheta)#

Therefore,

#costheta=2/sqrt5#

#sintheta=1/sqrt5#

#theta=0.46 rd#

#z=sqrt5(cos0.46+isin0.46)#

Therefore,

#z^5=(sqrt5(cos0.46+isin0.46))^5#

#=(sqrt5)^5(cos2.32+ isin2.32)#

#=55.9(-0.68+0.73i)#