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Beliebt Algebra >

z^8=1

Lösung

z^{8} = 1

Lösung

z = 1, z = - 1, z = i, z = - i, z = \frac{1}{2} + \frac{1}{2} i, z = - \frac{1}{2} - \frac{1}{2} i, z = \frac{1}{2} - \frac{1}{2} i, z = - \frac{1}{2} + \frac{1}{2} i

Schritte zur Lösung

z^{8} = 1

Schreibe die Gleichung um mit

u = z^{2}

und

u^{4} = z^{8}

u^{4} = 1

Löse

u^{4} = 1 : u = 1, u = - 1, u = i, u = - i

u = 1, u = - 1, u = i, u = - i

Setze

u = z^{2} w i e d er e in,

löse für

z

Löse

z^{2} = 1 : z = 1, z = - 1

Löse

z^{2} = - 1 : z = i, z = - i

Löse

z^{2} = i : z = \frac{1}{2} + \frac{1}{2} i, z = - \frac{1}{2} - \frac{1}{2} i

Löse

z^{2} = - i : z = \frac{1}{2} - \frac{1}{2} i, z = - \frac{1}{2} + \frac{1}{2} i

Die Lösungen sind

z = 1, z = - 1, z = i, z = - i, z = \frac{1}{2} + \frac{1}{2} i, z = - \frac{1}{2} - \frac{1}{2} i, z = \frac{1}{2} - \frac{1}{2} i, z = - \frac{1}{2} + \frac{1}{2} i

Graph

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