Exercise 425Algebra › Complex numbersHigh schoolAdded 3 Nov 2026

De Moivre's formula

#De Moivre#powers#trigonometric form

Statement

Compute $(1+i)^{10}$ using De Moivre's formula.

Hint

Write $1+i$ in trigonometric form first.

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Trigonometric form. $|1+i|=\sqrt2$ , $\arg(1+i)=\pi/4$ , so $1+i=\sqrt2 e^{i\pi/4}$ .

De Moivre. $(1+i)^{10}=(\sqrt2)^{10}e^{10i\pi/4}=2^5 e^{5i\pi/2}=32e^{i\pi/2}=32i$ .

Conclusion. $(1+i)^{10}=32i$ .

Verification. $(1+i)^2=2i$ , $(1+i)^4=-4$ , $(1+i)^8=16$ , $(1+i)^{10}=16\cdot(2i)=32i$ . Correct.