Power the length, multiply the angle: To raise a complex number to a power, switch to polar form and use De Moivre's theorem:
(r cisθ)ⁿ = rⁿ cis(nθ) — raise the modulus to the power, and multiply the argument by n.
Powers of cis(π/6): each power adds π/6 to the angle while staying on the unit circle.
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IB-style question — a high power
Find (1 + i)⁸.
Step by step
- Put 1 + i in polar form: r = √2, θ = π/4.
- Apply De Moivre: power the modulus, ×8 the angle.
- cis(2π) = 1 (a full turn).
Final answer
16.
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Why not just multiply it out?: Expanding (√3 + i)⁶ by hand would mean a huge binomial expansion. De Moivre turns it into one power and one multiplication — far faster and less error-prone.
IB-style question — sixth power
Find (√3 + i)⁶.
Step by step
- Polar form: r = √(3 + 1) = 2, θ = arctan(1/√3) = π/6.
- De Moivre with n = 6.
- cis(π) = −1.
Final answer
−64.