solvefor w,w^3=-2+2sqrt(3)i

Lösung

löse nach

w, w^{3} = - 2 + 23 i

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 + 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 = 3, a = - 2 + 23 i

∣ 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})

0 = 2 x^{3} - 10 x^{2} + x - 5

x (x + 5) (x + 7) = 0

so l v e f or x, 0 = x^{3} - x^{2} - 4 x + 4

x^{6} + 2 = 0

19 - 8 y^{6} = 5 y