De Moivre & powers

V1 — De Moivre: repeat polar multiply.

Builds on: polar multiply Tool (r’s multiply, θ’s add).

Tool gained: De Moivre — [r(cos θ + i sin θ)]ⁿ = rⁿ (cos(nθ) + i sin(nθ)).

1. Principle. Raising to n is multiplying the same move n times: r→rⁿ, θ→nθ.
2. Model. zⁿ = rⁿ (cos nθ + i sin nθ) for integer n (and carefully for rational).
3. Prediction / Answer. (√2∠π/4)⁴ = 4∠π = −4. Transfer: four equal phase steps around a circle.

V2 — Solve zⁿ = a.

Builds on: De Moivre (V1) — take nth root of r and split the angle.

Tool gained: n roots of a = R∠Θ are r=R^{1/n}, θ = (Θ+2πk)/n for k=0..n−1.

1. Principle. Undoing zⁿ undoes stretch with an nth root and undoes turn by dividing the angle — plus full turns 2πk that look the same before dividing.
2. Model. z_k = R^{1/n} [cos((Θ+2πk)/n) + i sin((Θ+2πk)/n)].
3. Prediction / Answer. Square roots of i: √2/2 · (1+i) and opposite. Transfer: n equal blades on a wheel.

V3 — Light Euler shorthand → middle Vortex next.

Builds on: polar cis form (polar Tool). Next climb: Euler / cis bridge is the middle Vortex (not a series leap).

Tool gained: light shorthand cis θ := cos θ + i sin θ, written e^{iθ} as notation only (bridge companion earns the working definition properly).

1. Principle. Once polar multiply is “add angles,” the same pattern as exp addition invites the shorthand e^{iθ} — a name for cis, not a free ticket into series.
2. Model. e^{iθ} := cos θ + i sin θ (working notation on this rung); De Moivre ↔ (e^{iθ})ⁿ = e^{inθ} as shorthand.
3. Prediction / Answer. e^{iπ} = −1 as cis π. Proper middle Vortex → Euler / cis bridge; full series proof stays an honest later Branch.