We start by writing z=-1+iz=−1+i, in trigonometric form
|z|=sqrt((-1)^2+1^2)=sqrt2|z|=√(−1)2+12=√2
z=sqrt2(-1/sqrt2+1/sqrt2i)z=√2(−1√2+1√2i)
costheta=-1/sqrt2cosθ=−1√2, =>⇒, theta=3/4piθ=34π
sintheta=1/sqrt2sinθ=1√2, =>⇒, theta=3/4piθ=34π
The trigonometric form is
z=sqrt2(cos(3/4pi)+isin(3/4pi))z=√2(cos(34π)+isin(34π))
Applying Demoivre's theorem
(costheta+isintheta)^n=cos(ntheta)+isin(ntheta)(cosθ+isinθ)n=cos(nθ)+isin(nθ)
z^4=(sqrt2(cos(3/4pi)+isin(3/4pi))^4z4=(√2(cos(34π)+isin(34π))4
=(sqrt2)^4(cos(3/4pi*4)+isin(3/4pi*4))=(√2)4(cos(34π⋅4)+isin(34π⋅4))
=4(cos(3pi)+isin(3pi))=4(cos(3π)+isin(3π))
=4(-1+i*0)=4(−1+i⋅0)
=-4=−4