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Beliebt Algebra >

x^5-32i=0

Lösung

x^{5} - 32 i = 0

Lösung

x = \frac{5 + 5}{2} + i \frac{3 - 5}{2}, x = 2 i, x = - \frac{5 + 5}{2} + i \frac{3 - 5}{2}, x = - \frac{10 - 2 5}{2} + i \frac{- 1 - 5}{2}, x = \frac{10 - 2 5}{2} + i \frac{- 1 - 5}{2}

Schritte zur Lösung

x^{5} - 32 i = 0

Verschiebe

32 i

auf die rechte Seite

x^{5} = 32 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 = 32 i

∣ a ∣ = 32

ar g (a) = \frac{π}{2}

x = 532 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{5})), x = 532 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{5})), x = 532 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{5})), x = 532 (cos (\frac{\frac{π}{2} + 2 \cdot 3 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 3 π}{5})), x = 532 (cos (\frac{\frac{π}{2} + 2 \cdot 4 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 4 π}{5}))

Vereinfache

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

Vereinfache

532 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{5})) : 2 i

Vereinfache

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

Vereinfache

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

Vereinfache

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

x = \frac{5 + 5}{2} + i \frac{3 - 5}{2}, x = 2 i, x = - \frac{5 + 5}{2} + i \frac{3 - 5}{2}, x = - \frac{10 - 2 5}{2} + i \frac{- 1 - 5}{2}, x = \frac{10 - 2 5}{2} + i \frac{- 1 - 5}{2}

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