De Moivre’s Theorem Calculator

De Moivre’s Theorem Formula

Worked Example

z = 1 + i with n = 3:

1. De Moivre: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)
2. Roots: r^(1/n)[cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)], k = 0 … n − 1

arg z (degrees): 45 · |zⁿ| = r^3: 2.828427125

GATE tip

For powers like (1 + i)¹⁰, convert to polar form first: √2∠45°, so the result is 32∠450° = 32i. This is much faster than repeated multiplication.

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Frequently asked questions

Sources

Last reviewed: 2026-10-15. Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.