De Moivre's theorem — powers
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Question
State De Moivre's theorem.
Answer
(r cisθ)ⁿ = rⁿ cis(nθ) — power the modulus, multiply the argument by n.
Question
How do you raise a complex number to a power?
Answer
Convert to polar form, apply De Moivre (rⁿ, nθ), then convert back if needed.
Question
Why convert to polar before powering?
Answer
De Moivre only applies to polar/exponential form; powering a + bi directly means a messy binomial expansion.
Question
(1 + i)⁸ = ?
Answer
(√2 cis(π/4))⁸ = (√2)⁸ cis(2π) = 16.
Question
(√3 + i)⁶ = ?
Answer
(2 cis(π/6))⁶ = 2⁶ cis(π) = −64.
Question
De Moivre in exponential form?
Answer
(r e^(iθ))ⁿ = rⁿ e^(inθ) — same idea via index laws.
Question
Common De Moivre mistake?
Answer
Multiplying r by n instead of powering it (it's rⁿ), or forgetting to multiply the angle by n.
Question
(1 − i)⁴ = ?
Answer
(√2 cis(−π/4))⁴ = 4 cis(−π) = −4.
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Full study notes for De Moivre's theorem — powers
Topic 1.14 hub
De Moivre & roots (HL only)
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