This Sounds Like Da-Movie Theory

Question

Use DeMoivre’s theorem to expand: (1 + i square root 3)⁴

Answer 1

(1 + i√3)⁴

(r(cosθ+isinθ))n=rn(cos(nθ)+isin(nθ)).

= 1 + 4 sqrt(3) i + 18 i^2 + 12 sqrt(3) i^3 + 9 i^4

Now let 1−√3i=r(cosθ+isinθ)

rcosθ=1 and rsinθ=−√3

hence squaring and adding r2=(12+(−√3)2)=1+3=4

and r=2, cosθ=12 and sinθ=−√32

hence, θ=−π3

And 1−√3i=2(cos(−π3)+isin(−π3)) and using DeMoivre's theorem

(1−√3i)3=23(cos(−3π3)+isin(−3π3))

= 23(cos(−π)+isin(−π))

= 8(−1+i⋅0)

= −8

Answer 2

DeMoivre's theorem tells us that multiplying a complex number by itself n times has the effect of multiplying its magnitude by itself n times and multiplying the angle it makes with the positive x axis by n. Therefore, we have to find the magnitude (r) and the angle (θ) of the given complex number (1+√3i) first:

r=√[1²+(√3)²]=√(1+3)=√4=2

θ=tan^-1(√3/1)=tan^-1(√3)=π/3

Since we are raising the given complex number to the 4^th power the new magnitude and angle of our complex number according to DeMoivre's theorem will be:

r⁴=2⁴=16

4θ=4π/3

Therefore, our new complex number can be expressed in polar form as follows:

r⁴[cos(4θ)+isin(4θ)]=16[cos(4π/3)+isin(4π/3)]

Since cos(4π/3)=-1/2 and sin(4π/3)=-√3/2 we can now express our complex number in standard form as follows:

16(-1/2-√3i/2)=-8-8√3i

EDIT: sin(4π/3)=-√3/2