How do you use DeMoivre's Theorem to simplify (√5−4i)3?

1 Answer
Shwetank Mauria
Nov 15, 2016

(√5−4i)3=−43√5−4i

Explanation:

According to DeMoivre's theorem

(r(cosθ+isinθ))n=rn(cosnθ+isinnθ)

Now let √5−4i=r(cosθ+isinθ)

hence rcosθ=1 and rsinθ=−√3

hence squaring and adding r2=((√5)2+(−4)2)=5+16=21

and r=√21, cosθ=√521 and sinθ=−4√21

Therefore using DeMoivre's theorem

(√5−4i)3=(√21)3(cos3θ+isin3θ)

As cos3θ=4cos3θ−3cosθ=4(√521)3−3√521

= 2021√521−3√521=−4321√521

and sin3θ=3sinθ−4sin3θ=3(−4√21)−4(−4√21)3

= −12√21+25621√21=−252+25621√21=−421√21

Hence, (√5−4i)3=(√21)3(−4321√521−i421√21)

= −43√5−4i

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