Computing (1+i)¹⁰

Complex Analysis · Axiom Academy

De Moivre's Theorem in action: powers of a complex number, the easy way. Compute (1+i)^ 10 . Expanding it directly would mean eleven binomial terms — instead we convert 1+i to polar form, apply De Moivre's Theorem, reduce the angle, and read off the answer. Nice work — you computed a 10th power without expanding a single binomial term. Convert first: , found from the modulus and the argument . De Moivre's Theorem: raising to the n th power raises the modulus to the n th power and multiplies the angle by n : and . Reduce the angle: , so the direction is straight up the imaginary axis. Polar form turns hard powers into easy arithmetic — scale the length, rotate the angle, reduce, and convert back.

This is the written version of the interactive lesson above. See the full Complex Analysis course.