How do you use DeMoivre's theorem to simplify (-1+i)^4(1+i)4?

1 Answer
Aug 21, 2017

The answer is =-4=4

Explanation:

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(12+12i)

costheta=-1/sqrt2cosθ=12, =>, theta=3/4piθ=34π

sintheta=1/sqrt2sinθ=12, =>, 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(34π4)+isin(34π4))

=4(cos(3pi)+isin(3pi))=4(cos(3π)+isin(3π))

=4(-1+i*0)=4(1+i0)

=-4=4