x^3=8i

Lösung

x^{3} = 8 i

x = 3 + i, x = - 3 + i, x = - 2 i

Schritte zur Lösung

x^{3} = 8 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 = 8 i

∣ a ∣ = 8

ar g (a) = \frac{π}{2}

x = 38 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{3})), x = 38 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{3})), x = 38 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{3}))

Vereinfache

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

Vereinfache

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

Vereinfache

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

x = 3 + i, x = - 3 + i, x = - 2 i

4 x^{3} = 0.2 5^{\frac{1}{2}}

so l v e f or x, 12 x^{5} = (4 x^{2}) (C)

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

so l v e f or y, y^{4} = x^{3} - 16 x

x^{6} - 2 x^{3} + 1 = 0