z^4=-2i

Lösung

z^{4} = - 2 i

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

Schritte zur Lösung

z^{4} = - 2 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 = - 2 i

∣ a ∣ = 2

ar g (a) = - \frac{π}{2}

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

Vereinfache

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

Vereinfache

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

Vereinfache

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

Vereinfache

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

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

so l v e f or y, x = y^{3}

x^{3} - 4 x^{2} + 2 x - 8 = 0

x^{4} - 9 x^{2} - 4 x = 1, 2 = 0

(x - 2)^{2} (1 - x) = - 2

1000 (\frac{1}{2}) \cdot x^{28} = 20