De Moivre's Theorem states that for complex number
#z = r(costheta + isintheta)#
#z^n = r^n(cos(ntheta) + isin(ntheta))#
So we need to get our complex number into modulus-argument form.
For #z = x+yi#
#r= sqrt(x^2+y^2) and theta = tan^(-1)(y/x) " (usually!)"#
I say usually because the number may be in a different quadrant and require some action.
#r=sqrt(1^2+1^2) = sqrt(2)#
#theta = tan^(-1)((1)/(-1)) = pi - tan^(-1)(1) = (3pi)/4#
So #z = sqrt(2)(cos((3pi)/4) + isin((3pi)/4))#
#z^(11) = (sqrt(2))^11(cos((33pi)/4) + isin((33pi)/4))#
#z^11 = 2^(11/2)(cos((pi)/4) + isin((pi)/4))#
#z^11 = 2^(11/2)(1/(sqrt(2)) + 1/(sqrt(2))i) = 2^(11/2)(2^(-1/2) + 2^(-1/2)i)#
#z^11 = 2^(11/2-1/2) + 2^(11/2-1/2)i = 2^5 + 2^5i#
#z^11 = 32 + 32i#
