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Beliebt Algebra >

z^6=i

Lösung

z^{6} = i

Lösung

z = \frac{2 ( 1 + 3 )}{4} + i \frac{2 ( - 1 + 3 )}{4}, z = \frac{2 ( - 1 + 3 )}{4} + i \frac{2 ( 1 + 3 )}{4}, z = - \frac{2}{2} + i \frac{2}{2}, z = \frac{2 ( - 1 - 3 )}{4} + i \frac{2 ( 1 - 3 )}{4}, z = \frac{2 ( 1 - 3 )}{4} + i \frac{2 ( - 1 - 3 )}{4}, z = \frac{2}{2} - i \frac{2}{2}

Schritte zur Lösung

z^{6} = 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 = 6, a = i

∣ a ∣ = 1

ar g (a) = \frac{π}{2}

z = 61 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{6}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{6})), z = 61 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{6}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{6})), z = 61 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{6}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{6})), z = 61 (cos (\frac{\frac{π}{2} + 2 \cdot 3 π}{6}) + i sin (\frac{\frac{π}{2} + 2 \cdot 3 π}{6})), z = 61 (cos (\frac{\frac{π}{2} + 2 \cdot 4 π}{6}) + i sin (\frac{\frac{π}{2} + 2 \cdot 4 π}{6})), z = 61 (cos (\frac{\frac{π}{2} + 2 \cdot 5 π}{6}) + i sin (\frac{\frac{π}{2} + 2 \cdot 5 π}{6}))

Vereinfache

61 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{6}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{6})) : \frac{2 ( 1 + 3 )}{4} + i \frac{2 ( - 1 + 3 )}{4}

Vereinfache

61 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{6}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{6})) : \frac{2 ( - 1 + 3 )}{4} + i \frac{2 ( 1 + 3 )}{4}

Vereinfache

61 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{6}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{6})) : - \frac{2}{2} + i \frac{2}{2}

Vereinfache

61 (cos (\frac{\frac{π}{2} + 2 \cdot 3 π}{6}) + i sin (\frac{\frac{π}{2} + 2 \cdot 3 π}{6})) : \frac{2 ( - 1 - 3 )}{4} + i \frac{2 ( 1 - 3 )}{4}

Vereinfache

61 (cos (\frac{\frac{π}{2} + 2 \cdot 4 π}{6}) + i sin (\frac{\frac{π}{2} + 2 \cdot 4 π}{6})) : \frac{2 ( 1 - 3 )}{4} + i \frac{2 ( - 1 - 3 )}{4}

Vereinfache

61 (cos (\frac{\frac{π}{2} + 2 \cdot 5 π}{6}) + i sin (\frac{\frac{π}{2} + 2 \cdot 5 π}{6})) : \frac{2}{2} - i \frac{2}{2}

z = \frac{2 ( 1 + 3 )}{4} + i \frac{2 ( - 1 + 3 )}{4}, z = \frac{2 ( - 1 + 3 )}{4} + i \frac{2 ( 1 + 3 )}{4}, z = - \frac{2}{2} + i \frac{2}{2}, z = \frac{2 ( - 1 - 3 )}{4} + i \frac{2 ( 1 - 3 )}{4}, z = \frac{2 ( 1 - 3 )}{4} + i \frac{2 ( - 1 - 3 )}{4}, z = \frac{2}{2} - i \frac{2}{2}

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