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Beliebt Algebra >

x^{16}-1=0

Lösung

x^{16} - 1 = 0

Lösung

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

Schritte zur Lösung

x^{16} - 1 = 0

Verschiebe

1

auf die rechte Seite

x^{16} = 1

Schreibe die Gleichung um mit

u = x^{2}

und

u^{8} = x^{16}

u^{8} = 1

Löse

u^{8} = 1 : u = 1, u = - 1, u = i, u = - 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 = 1, u = - 1, u = i, u = - 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 = x^{2} w i e d er e in,

löse für

x

Löse

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

Löse

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

Löse

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

Löse

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

Löse

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

Löse

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

Löse

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

Löse

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

Die Lösungen sind

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

Graph

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