Raise a complex number to an integer power using De Moivre's Theorem: [r(cos θ + i sin θ)]n = rn(cos nθ + i sin nθ). Enter z in polar or rectangular form, choose an exponent, and see the full computation with results in both polar and rectangular form, plus an interactive Argand diagram showing each successive power.
[r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)
Any complex number can be written in polar form: z = r·cis(θ) where r = |z| is the modulus and θ is the argument (angle). Multiplying two complex numbers in polar form multiplies their moduli and adds their angles: z₁·z₂ = r₁r₂·cis(θ₁+θ₂).
Applying this rule n times gives De Moivre's Theorem: the modulus is raised to the nth power and the angle is multiplied by n. When r = 1, every power of z lies on the unit circle — z just rotates by θ each step.
Computing powers efficiently
Instead of multiplying (a+bi)·(a+bi)·… n times, convert to polar, apply De Moivre, and convert back. Far fewer steps for large n.
Proving trig identities
Expanding (cos θ + i sin θ)ⁿ with the Binomial Theorem and matching real/imaginary parts gives exact formulas for cos(nθ) and sin(nθ) in terms of powers of cos θ and sin θ.
Finding nth roots
The inverse problem — finding all z such that zⁿ = w — uses De Moivre in reverse to produce n equally-spaced roots on a circle of radius r^(1/n). See the nth roots calculator below.
Our tutors can walk you through De Moivre's Theorem, polar form, and trig identities step by step — at your pace.