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{4 π}{9}) + 34 i sin (\frac{4 π}{9}), w = 34 cos (\frac{10 π}{9}) + 34 i sin (\frac{10 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{4 π}{9}) + 34 i sin (\frac{4 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{10 π}{9}) + 34 i sin (\frac{10 π}{9})

w = 34 cos (\frac{2 π}{9}) - 34 i sin (\frac{2 π}{9}), w = 34 cos (\frac{4 π}{9}) + 34 i sin (\frac{4 π}{9}), w = 34 cos (\frac{10 π}{9}) + 34 i sin (\frac{10 π}{9})

(x^{2} - x)^{2} - (x^{2} - x) - 12 = 0

so l v e f or x, y = x^{5}

x^{3} - 7 x^{2} + 5 x + 6 = 0

3 y^{4} - 5 y^{3} + 2 y^{2} = 0

x^{5} + x^{2} = 1