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One multiplication instead of a hundred

Your guide: Sir AhmedSays brackets have saved more marks than any calculator.

The problem

Squaring 3 + 4i by hand is fine — expand the brackets once and you're done. But raising √3 + i to the 100th power that way would take a lesson on its own.

Polar form skips all of that. Convert to r and θ just once, and De Moivre's theorem turns any power into a single multiplication of the angle.

For z = r(cosθ + isinθ), zⁿ = rⁿ(cos nθ + isin nθ). So the real part of zⁿ is rⁿcos nθ, and the imaginary part is rⁿsin nθ. A quotient (z1/z2)ⁿ works the same way, using (r1/r2)ⁿ and the angle n(θ1 − θ2).

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