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Beliebt Algebra >

z^3=-2+2i

Lösung

z^{3} = - 2 + 2 i

Lösung

z = 1 + i, z = \frac{- 1 - 3}{2} + i \frac{- 1 + 3}{2}, z = \frac{- 1 + 3}{2} + i \frac{- 1 - 3}{2}

Schritte zur Lösung

z^{3} = - 2 + 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 = 3, a = - 2 + 2 i

∣ a ∣ = 22

ar g (a) = - \frac{π}{4} + π

z = 322 (cos (\frac{- \frac{π}{4} + π + 2 \cdot 0 π}{3}) + i sin (\frac{- \frac{π}{4} + π + 2 \cdot 0 π}{3})), z = 322 (cos (\frac{- \frac{π}{4} + π + 2 \cdot 1 π}{3}) + i sin (\frac{- \frac{π}{4} + π + 2 \cdot 1 π}{3})), z = 322 (cos (\frac{- \frac{π}{4} + π + 2 \cdot 2 π}{3}) + i sin (\frac{- \frac{π}{4} + π + 2 \cdot 2 π}{3}))

Vereinfache

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

Vereinfache

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

Vereinfache

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

z = 1 + i, z = \frac{- 1 - 3}{2} + i \frac{- 1 + 3}{2}, z = \frac{- 1 + 3}{2} + i \frac{- 1 - 3}{2}

Beliebte Beispiele

x^{4} + 9 x^{2} + 20 = 0

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2 x^{3} + 25 = x^{2} + 50 x

2 x^{3} - 3 x^{2} = 6 x - 9

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