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Beliebt Algebra >

z^5=-3i

Lösung

z^{5} = - 3 i

Lösung

z = \frac{5 3 2 5 + 5}{4} - i \frac{5 3 2 3 - 5}{4}, z = \frac{5 3 10 - 2 5}{4} + i \frac{5 3 + 5 3 5}{4}, z = - \frac{5 3 10 - 2 5}{4} + i \frac{5 3 + 5 3 5}{4}, z = - \frac{5 3 2 5 + 5}{4} - i \frac{5 3 2 3 - 5}{4}, z = - 53 i

Schritte zur Lösung

z^{5} = - 3 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 = 5, a = - 3 i

∣ a ∣ = 3

ar g (a) = - \frac{π}{2}

z = 53 (cos (\frac{- \frac{π}{2} + 2 \cdot 0 π}{5}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 0 π}{5})), z = 53 (cos (\frac{- \frac{π}{2} + 2 \cdot 1 π}{5}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 1 π}{5})), z = 53 (cos (\frac{- \frac{π}{2} + 2 \cdot 2 π}{5}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 2 π}{5})), z = 53 (cos (\frac{- \frac{π}{2} + 2 \cdot 3 π}{5}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 3 π}{5})), z = 53 (cos (\frac{- \frac{π}{2} + 2 \cdot 4 π}{5}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 4 π}{5}))

Vereinfache

53 (cos (\frac{- \frac{π}{2} + 2 \cdot 0 π}{5}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 0 π}{5})) : \frac{5 3 2 5 + 5}{4} - i \frac{5 3 2 3 - 5}{4}

Vereinfache

53 (cos (\frac{- \frac{π}{2} + 2 \cdot 1 π}{5}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 1 π}{5})) : \frac{5 3 10 - 2 5}{4} + i \frac{5 3 + 5 3 5}{4}

Vereinfache

53 (cos (\frac{- \frac{π}{2} + 2 \cdot 2 π}{5}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 2 π}{5})) : - \frac{5 3 10 - 2 5}{4} + i \frac{5 3 + 5 3 5}{4}

Vereinfache

53 (cos (\frac{- \frac{π}{2} + 2 \cdot 3 π}{5}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 3 π}{5})) : - \frac{5 3 2 5 + 5}{4} - i \frac{5 3 2 3 - 5}{4}

Vereinfache

53 (cos (\frac{- \frac{π}{2} + 2 \cdot 4 π}{5}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 4 π}{5})) : - 53 i

z = \frac{5 3 2 5 + 5}{4} - i \frac{5 3 2 3 - 5}{4}, z = \frac{5 3 10 - 2 5}{4} + i \frac{5 3 + 5 3 5}{4}, z = - \frac{5 3 10 - 2 5}{4} + i \frac{5 3 + 5 3 5}{4}, z = - \frac{5 3 2 5 + 5}{4} - i \frac{5 3 2 3 - 5}{4}, z = - 53 i

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