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Beliebt Algebra >

z^8+1=0

Lösung

z^{8} + 1 = 0

Lösung

z = \frac{1}{2 ^{\frac{3}{4}} 2 - 1} + \frac{2 - 1}{4 2 ^{3}} i, z = - \frac{1}{2 ^{\frac{3}{4}} 2 - 1} - \frac{2 - 1}{4 2 ^{3}} i, z = - \frac{1}{2 ^{\frac{3}{4}} 1 + 2} + \frac{1 + 2}{4 2 ^{3}} i, z = \frac{1}{2 ^{\frac{3}{4}} 1 + 2} - \frac{1 + 2}{4 2 ^{3}} i, z = \frac{1}{2 ^{\frac{3}{4}} 1 + 2} + \frac{1 + 2}{4 2 ^{3}} i, z = - \frac{1}{2 ^{\frac{3}{4}} 1 + 2} - \frac{1 + 2}{4 2 ^{3}} i, z = - \frac{1}{2 ^{\frac{3}{4}} 2 - 1} + \frac{2 - 1}{4 2 ^{3}} i, z = \frac{1}{2 ^{\frac{3}{4}} 2 - 1} - \frac{2 - 1}{4 2 ^{3}} i

Schritte zur Lösung

z^{8} + 1 = 0

Verschiebe

1

auf die rechte Seite

z^{8} = - 1

Schreibe die Gleichung um mit

u = z^{2}

und

u^{4} = z^{8}

u^{4} = - 1

Löse

u^{4} = - 1 : u = \frac{1}{2} + \frac{1}{2} i, u = - \frac{1}{2} - \frac{1}{2} i, u = - \frac{1}{2} + \frac{1}{2} i, u = \frac{1}{2} - \frac{1}{2} i

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

Setze

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

löse für

z

Löse

z^{2} = \frac{1}{2} + \frac{1}{2} i : z = \frac{1}{2 ^{\frac{3}{4}} 2 - 1} + \frac{2 - 1}{4 2 ^{3}} i, z = - \frac{1}{2 ^{\frac{3}{4}} 2 - 1} - \frac{2 - 1}{4 2 ^{3}} i

Löse

z^{2} = - \frac{1}{2} - \frac{1}{2} i : z = - \frac{1}{2 ^{\frac{3}{4}} 1 + 2} + \frac{1 + 2}{4 2 ^{3}} i, z = \frac{1}{2 ^{\frac{3}{4}} 1 + 2} - \frac{1 + 2}{4 2 ^{3}} i

Löse

z^{2} = - \frac{1}{2} + \frac{1}{2} i : z = \frac{1}{2 ^{\frac{3}{4}} 1 + 2} + \frac{1 + 2}{4 2 ^{3}} i, z = - \frac{1}{2 ^{\frac{3}{4}} 1 + 2} - \frac{1 + 2}{4 2 ^{3}} i

Löse

z^{2} = \frac{1}{2} - \frac{1}{2} i : z = - \frac{1}{2 ^{\frac{3}{4}} 2 - 1} + \frac{2 - 1}{4 2 ^{3}} i, z = \frac{1}{2 ^{\frac{3}{4}} 2 - 1} - \frac{2 - 1}{4 2 ^{3}} i

Die Lösungen sind

z = \frac{1}{2 ^{\frac{3}{4}} 2 - 1} + \frac{2 - 1}{4 2 ^{3}} i, z = - \frac{1}{2 ^{\frac{3}{4}} 2 - 1} - \frac{2 - 1}{4 2 ^{3}} i, z = - \frac{1}{2 ^{\frac{3}{4}} 1 + 2} + \frac{1 + 2}{4 2 ^{3}} i, z = \frac{1}{2 ^{\frac{3}{4}} 1 + 2} - \frac{1 + 2}{4 2 ^{3}} i, z = \frac{1}{2 ^{\frac{3}{4}} 1 + 2} + \frac{1 + 2}{4 2 ^{3}} i, z = - \frac{1}{2 ^{\frac{3}{4}} 1 + 2} - \frac{1 + 2}{4 2 ^{3}} i, z = - \frac{1}{2 ^{\frac{3}{4}} 2 - 1} + \frac{2 - 1}{4 2 ^{3}} i, z = \frac{1}{2 ^{\frac{3}{4}} 2 - 1} - \frac{2 - 1}{4 2 ^{3}} i

Graph

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