Write the complex number z 1 squareroot 3 i19 in polar for

Write the complex number z = (1 + squareroot 3 i)^19 in polar form: z = r(cos theta + i sin theta) where r = 524288 and theta = 1pi The angle should satisfy 0 lessthanorequalto theta

Solution

(1+sqrt3 i)¹⁹

r=sqrt(1+3)=2

theta=tan^-1(sqrt3)=pi/3

(2(cos(pi/3)+isin(pi/3)))¹⁹

2¹⁹(cos(19pi/3)+isin(19pi/3))

524288(cos(19pi/3)+isin(19pi/3))

r=524488

and theta=19pi/3