De Moivre's Theorem Calculator
Calculate powers of complex numbers using De Moivre's theorem: (r·cis(θ))^n = r^n·cis(n·θ).
Complex Number (Polar Form)
Original number:
1.7321 + 1.0000i
2 ∠ 30.00°
z^3
0.0000 + 8.0000i
8.0000 ∠ 90.00°
Step by Step
z = 2(cos 30.00° + i·sin 30.00°)
z^3 = 2^3 · (cos(3·30.00°) + i·sin(3·30.00°))
z^3 = 8.0000 · (cos 90.00° + i·sin 90.00°)
z^3 = 0.0000 + 8.0000i
3th Roots of z
k=0: 1.2408 + 0.2188i(1.2599 ∠ 10.00°)
k=1: -0.8099 + 0.9652i(1.2599 ∠ 130.00°)
k=2: -0.4309 - 1.1839i(1.2599 ∠ 250.00°)
De Moivre's Theorem
Powers
[r(cos θ + i sin θ)]^n
= r^n (cos(nθ) + i sin(nθ))
nth Roots
z^(1/n) = r^(1/n) · cis((θ + 2πk)/n)
for k = 0, 1, 2, ..., n-1