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Beliebt Algebra >

z^{12}=1

Lösung

z^{12} = 1

Lösung

z = 1, z = - 1, z = i, z = - i, z = \frac{3}{2} + \frac{1}{2} i, z = - \frac{3}{2} - \frac{1}{2} i, z = - \frac{1}{2} + \frac{3}{2} i, z = \frac{1}{2} - \frac{3}{2} i, z = \frac{1}{2} + \frac{3}{2} i, z = - \frac{1}{2} - \frac{3}{2} i, z = - \frac{3}{2} + \frac{1}{2} i, z = \frac{3}{2} - \frac{1}{2} i

Schritte zur Lösung

z^{12} = 1

Schreibe die Gleichung um mit

u = z^{2}

und

u^{6} = z^{12}

u^{6} = 1

Löse

u^{6} = 1 : u = 1, u = - 1, u = \frac{1}{2} + \frac{3}{2} i, u = - \frac{1}{2} - \frac{3}{2} i, u = - \frac{1}{2} + \frac{3}{2} i, u = \frac{1}{2} - \frac{3}{2} i

u = 1, u = - 1, u = \frac{1}{2} + \frac{3}{2} i, u = - \frac{1}{2} - \frac{3}{2} i, u = - \frac{1}{2} + \frac{3}{2} i, u = \frac{1}{2} - \frac{3}{2} 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} = \frac{1}{2} + \frac{3}{2} i : z = \frac{3}{2} + \frac{1}{2} i, z = - \frac{3}{2} - \frac{1}{2} i

Löse

z^{2} = - \frac{1}{2} - \frac{3}{2} i : z = - \frac{1}{2} + \frac{3}{2} i, z = \frac{1}{2} - \frac{3}{2} i

Löse

z^{2} = - \frac{1}{2} + \frac{3}{2} i : z = \frac{1}{2} + \frac{3}{2} i, z = - \frac{1}{2} - \frac{3}{2} i

Löse

z^{2} = \frac{1}{2} - \frac{3}{2} i : z = - \frac{3}{2} + \frac{1}{2} i, z = \frac{3}{2} - \frac{1}{2} i

Die Lösungen sind

z = 1, z = - 1, z = i, z = - i, z = \frac{3}{2} + \frac{1}{2} i, z = - \frac{3}{2} - \frac{1}{2} i, z = - \frac{1}{2} + \frac{3}{2} i, z = \frac{1}{2} - \frac{3}{2} i, z = \frac{1}{2} + \frac{3}{2} i, z = - \frac{1}{2} - \frac{3}{2} i, z = - \frac{3}{2} + \frac{1}{2} i, z = \frac{3}{2} - \frac{1}{2} i

Graph

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