z^3=27i

Lösung

z^{3} = 27 i

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

Schritte zur Lösung

z^{3} = 27 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 = 27 i

∣ a ∣ = 27

ar g (a) = \frac{π}{2}

z = 327 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{3})), z = 327 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{3})), z = 327 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{3}))

Vereinfache

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

Vereinfache

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

Vereinfache

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

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

(x^{2} + 4) (x + 1) = 0

0 = x^{3} - 4 x^{2} - 9 x + 36

x^{3} + 4 x^{2} - 4 x - 16 = 0

x^{3} + 2 x^{2} - 16 x - 32 = 0

x^{3} = - 512