w^3=-2+2isqrt(3)

Lösung

w^{3} = - 2 + 2 i 3

w = 34 cos (\frac{2 π}{9}) + 34 i sin (\frac{2 π}{9}), w = 34 cos (\frac{8 π}{9}) + 34 i sin (\frac{8 π}{9}), w = 34 cos (\frac{14 π}{9}) + 34 i sin (\frac{14 π}{9})

Schritte zur Lösung

w^{3} = - 2 + 2 i 3

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 3

∣ a ∣ = 4

ar g (a) = - \frac{π}{3} + π

w = 34 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 0 π}{3}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 0 π}{3})), w = 34 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 1 π}{3}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 1 π}{3})), w = 34 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 2 π}{3}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 2 π}{3}))

Vereinfache

34 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 0 π}{3}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 0 π}{3})) : 34 cos (\frac{2 π}{9}) + 34 i sin (\frac{2 π}{9})

Vereinfache

34 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 1 π}{3}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 1 π}{3})) : 34 cos (\frac{8 π}{9}) + 34 i sin (\frac{8 π}{9})

Vereinfache

34 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 2 π}{3}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 2 π}{3})) : 34 cos (\frac{14 π}{9}) + 34 i sin (\frac{14 π}{9})

w = 34 cos (\frac{2 π}{9}) + 34 i sin (\frac{2 π}{9}), w = 34 cos (\frac{8 π}{9}) + 34 i sin (\frac{8 π}{9}), w = 34 cos (\frac{14 π}{9}) + 34 i sin (\frac{14 π}{9})

y^{3} - 6 y^{2} + 8 y = 0

w^{4} = - 1

5 x^{3} + 40 x^{2} + 80 x = 0

48 x^{3} + 32 x^{2} - 75 x - 50 = 0

r^{3} = - 1