z^4=-2+2sqrt(3)i

Lösung

z^{4} = - 2 + 23 i

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

Schritte zur Lösung

z^{4} = - 2 + 23 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 + 23 i

∣ a ∣ = 4

ar g (a) = - \frac{π}{3} + π

z = 44 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 0 π}{4}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 0 π}{4})), z = 44 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 1 π}{4}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 1 π}{4})), z = 44 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 2 π}{4}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 2 π}{4})), z = 44 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 3 π}{4}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 3 π}{4}))

Vereinfache

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

Vereinfache

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

Vereinfache

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

Vereinfache

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

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

x (x + 7) (x^{2} + 49) = 0

so l v e f or x, x^{3} + y^{3} = - 9

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

x^{3} + 3 x^{2} + 9 x + 27 = 0

9 x^{4} - 9 x^{2} + 2 = 0