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Beliebt Algebra >

x^5=i

Lösung

x^{5} = i

Lösung

x = \frac{2 5 + 5}{4} + i \frac{2 3 - 5}{4}, x = i, x = - \frac{2 5 + 5}{4} + i \frac{2 3 - 5}{4}, x = - \frac{10 - 2 5}{4} + i \frac{- 1 - 5}{4}, x = \frac{10 - 2 5}{4} + i \frac{- 1 - 5}{4}

Schritte zur Lösung

x^{5} = 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 = i

∣ a ∣ = 1

ar g (a) = \frac{π}{2}

x = 51 (cos (\frac{\frac{π}{2} + 2 \cdot 0 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 0 π}{5})), x = 51 (cos (\frac{\frac{π}{2} + 2 \cdot 1 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 1 π}{5})), x = 51 (cos (\frac{\frac{π}{2} + 2 \cdot 2 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 2 π}{5})), x = 51 (cos (\frac{\frac{π}{2} + 2 \cdot 3 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 3 π}{5})), x = 51 (cos (\frac{\frac{π}{2} + 2 \cdot 4 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \cdot 4 π}{5}))

Vereinfache

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

Vereinfache

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

Vereinfache

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

Vereinfache

51 (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}{4} + i \frac{- 1 - 5}{4}

Vereinfache

51 (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}{4} + i \frac{- 1 - 5}{4}

x = \frac{2 5 + 5}{4} + i \frac{2 3 - 5}{4}, x = i, x = - \frac{2 5 + 5}{4} + i \frac{2 3 - 5}{4}, x = - \frac{10 - 2 5}{4} + i \frac{- 1 - 5}{4}, x = \frac{10 - 2 5}{4} + i \frac{- 1 - 5}{4}

Beliebte Beispiele

r^5+5r^4-2r^3-10r^2+r+5=0

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