Let z4=81i, let z=reiθ=r(cosθ+isinθ). [If z=a+ib, a=rcosθ, b=rsinθ.]
z4=r4e4iθ=r4(cos(4θ)+isin(4θ))=81i.
r4=81, r=3; cos(4θ)+isin(4θ)=i, 4θ=(4n+1)π/2, θ=(4n+1)π/8 radians or (4n+1)22.5°.
z=3(cos((4n+1)π/8)+isin((4n+1)π/8).
n=0: z=3(cos(π/8)+isin(π/8));
n=1: z=3(cos(5π/8)+isin(3π/8));
n=2: z=3(cos(9π/8)+isin(5π/8));
n=3: z=3(cos(13π/8)+isin(7π/8));
Solution: z=2.7716386+1.1480503i, -1.1480503+2.7716386i, -2.7716386-1.1480503i, 1.1480503-2.7716386i approximately.
[cos(π/4)=2cos2(π/8)-1, cos2(π/8)=½(1+cos(π/4)), cos(π/8)=√(½(1+cos(π/4)))=√(½(1+√2/2)); sin2(π/8)=1-½(1+cos(π/4))=½(1-√2/2), sin(π/8)=√(½(1-√2/2)).]
ARGAND DIAGRAM
The four zeroes are shown as the points A, B, C, D. They form the sides of a square (blue), and the semi-diagonals (red) have a length (magnitude) of 3 showing that all the zeroes have magnitude 3 (4th root of 81), The green circle merely emphasises the constant magnitude. Note that the diagonals are inclined at an angle 22.5° (π/8).