z^4=1-i

Lösung

z^{4} = 1 - i

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}

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

Vereinfache

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

5 x^{3} + x = 0

x^{3} - 10 x^{2} + 27 x - 18 = 0

(1 - x) (x + 2) 2 (x + 3) = 0

- λ^{3} + 6 λ^{2} - 11 λ + 6 = 0

x^{3} - x^{2} - 4 x = - 4