`(2 + 2i)^6` Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form.

`z=(2+2i)^6`

`r=sqrt[(2)^2+(2)^2]=sqrt8=2sqrt2`

`theta=arctan(2/2)=arctan(1)=pi/4`

DeMoivre's Theorem

`z^n=[r(costheta+isintheta)]^n=r^n(cosntheta+isinntheta)`

`z^6=[2sqrt2(cos(pi/4)+isin(pi/4)]^6=(2sqrt2)^6[cos6(pi/4)+isin6(pi/4)]`

`z^6=512[cos((3pi)/2)+isin((3pi)/2)]`

`z^6=512[0+(-1)i]`

`z^6=-512i`

`z=(2+2i)^6`

`r=sqrt[(2)^2+(2)^2]=sqrt8=2sqrt2`

`theta=arctan(2/2)=arctan(1)=pi/4`

DeMoivre's Theorem

`z^n=[r(costheta+isintheta)]^n=r^n(cosntheta+isinntheta)`

`z^6=[2sqrt2(cos(pi/4)+isin(pi/4)]^6=(2sqrt2)^6[cos6(pi/4)+isin6(pi/4)]`

`z^6=512[cos((3pi)/2)+isin((3pi)/2)]`

`z^6=512[0+(-1)i]`

`z^6=-512i`