`2(sqrt(3) + i)^10` Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form.

Given `2(sqrt3+i)^10`

Let `z=(sqrt3+i)^10`

`r=sqrt[(sqrt3)^2+(1)^2]=sqrt(3+1)=sqrt4=2`

`theta=arctan(1/sqrt3)=pi/6`

DeMoivre's Theorem

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

`z^10=[2(cos(pi/6)+isin(pi/6))]^10`

`z^10=2^10(cos10(pi/6)+isin10(pi/6))`

`z^10=1024[cos((5pi)/3+isin((5pi)/3)]`

`z^10=1024[1/2+(-sqrt3/2)i]`

`z^10=512-512sqrt3i`

`2z=2(512-512sqrt3i)=1024-1024sqrt3i`