(1 + i)20
Raising a complex number to a power is generally easiest if you write the complex number in r·cis(theta) form.
For the complex number x + yi:
r = sqrt( x2 + y2 ) = sqrt( 12 + 12 ) = sqrt(2)
theta = tan-1( y/x ) = tan-1( 1/1 ) = tan-1( 1) = 45o. (you can also use radians)
Therefore, 1 + i = sqrt(2) · cis( 45o )
Formula: (x + yi)n = [ r·cis(theta) ]n = rn · cis( n · theta )
(1 + i)20 = [ sqrt(2) · cis( 45o ) ]20 = [sqrt(2)]20 · cis( 20 · 45o ) = 1024·cis( 900o ) = 1024·cis( 180o )
= 1024·cos(180o) + 1024·i·sin(180o) = 1024·(-1) + 1024·i·(0) = -1024 + 0i