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Beliebt Algebra >

z^4=-1+i

Lösung

z^{4} = - 1 + i

Lösung

z = \frac{8 2 2 + 2 - 2}{2} + i \frac{8 2 2 - 2 - 2}{2}, z = - \frac{8 2 2 - 2 - 2}{2} + i \frac{8 2 2 + 2 - 2}{2}, z = - \frac{8 2 2 + 2 - 2}{2} - i \frac{8 2 2 - 2 - 2}{2}, z = \frac{8 2 2 - 2 - 2}{2} - i \frac{8 2 2 + 2 - 2}{2}

Schritte zur Lösung

z^{4} = - 1 + i

Für

z^{n} = a

sind die Lösungen

z_{k} = n ∣ a ∣ (cos (\frac{ar g ( a ) + 2 kπ}{n}) + i sin (\frac{ar g ( a ) + 2 kπ}{n})),

k = 0, 1, \dots, n - 1

Für

n = 4, a = - 1 + i

∣ a ∣ = 2

ar g (a) = - \frac{π}{4} + π

z = 42 (cos (\frac{- \frac{π}{4} + π + 2 \cdot 0 π}{4}) + i sin (\frac{- \frac{π}{4} + π + 2 \cdot 0 π}{4})), z = 42 (cos (\frac{- \frac{π}{4} + π + 2 \cdot 1 π}{4}) + i sin (\frac{- \frac{π}{4} + π + 2 \cdot 1 π}{4})), z = 42 (cos (\frac{- \frac{π}{4} + π + 2 \cdot 2 π}{4}) + i sin (\frac{- \frac{π}{4} + π + 2 \cdot 2 π}{4})), z = 42 (cos (\frac{- \frac{π}{4} + π + 2 \cdot 3 π}{4}) + i sin (\frac{- \frac{π}{4} + π + 2 \cdot 3 π}{4}))

Vereinfache

42 (cos (\frac{- \frac{π}{4} + π + 2 \cdot 0 π}{4}) + i sin (\frac{- \frac{π}{4} + π + 2 \cdot 0 π}{4})) : \frac{8 2 2 + 2 - 2}{2} + i \frac{8 2 2 - 2 - 2}{2}

Vereinfache

42 (cos (\frac{- \frac{π}{4} + π + 2 \cdot 1 π}{4}) + i sin (\frac{- \frac{π}{4} + π + 2 \cdot 1 π}{4})) : - \frac{8 2 2 - 2 - 2}{2} + i \frac{8 2 2 + 2 - 2}{2}

Vereinfache

42 (cos (\frac{- \frac{π}{4} + π + 2 \cdot 2 π}{4}) + i sin (\frac{- \frac{π}{4} + π + 2 \cdot 2 π}{4})) : - \frac{8 2 2 + 2 - 2}{2} - i \frac{8 2 2 - 2 - 2}{2}

Vereinfache

42 (cos (\frac{- \frac{π}{4} + π + 2 \cdot 3 π}{4}) + i sin (\frac{- \frac{π}{4} + π + 2 \cdot 3 π}{4})) : \frac{8 2 2 - 2 - 2}{2} - i \frac{8 2 2 + 2 - 2}{2}

z = \frac{8 2 2 + 2 - 2}{2} + i \frac{8 2 2 - 2 - 2}{2}, z = - \frac{8 2 2 - 2 - 2}{2} + i \frac{8 2 2 + 2 - 2}{2}, z = - \frac{8 2 2 + 2 - 2}{2} - i \frac{8 2 2 - 2 - 2}{2}, z = \frac{8 2 2 - 2 - 2}{2} - i \frac{8 2 2 + 2 - 2}{2}

Beliebte Beispiele

2 x^{5} + 24 x = 14 x^{3}

1/2 =4x^3-9x^2-1

\frac{1}{2} = 4 x^{3} - 9 x^{2} - 1

2 x^{4} - 2 = 0

x^{6} + 3 x^{2} = 0

- x^{4} + 1 = 0