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Beliebt Algebra >

z^4=-7+24i

Lösung

z^{4} = - 7 + 24 i

Lösung

z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + π}{4}), z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + 3 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 3 π}{4}), z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + 5 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 5 π}{4}), z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + 7 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 7 π}{4})

Schritte zur Lösung

z^{4} = - 7 + 24 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 = 4, a = - 7 + 24 i

∣ a ∣ = 25

ar g (a) = - arctan (\frac{24}{7}) + π

z = 425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 0 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 0 π}{4})), z = 425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 1 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 1 π}{4})), z = 425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 2 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 2 π}{4})), z = 425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 3 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 3 π}{4}))

Vereinfache

425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 0 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 0 π}{4})) : 5 cos (\frac{- arctan ( \frac{24}{7} ) + π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + π}{4})

Vereinfache

425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 1 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 1 π}{4})) : 5 cos (\frac{- arctan ( \frac{24}{7} ) + 3 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 3 π}{4})

Vereinfache

425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 2 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 2 π}{4})) : 5 cos (\frac{- arctan ( \frac{24}{7} ) + 5 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 5 π}{4})

Vereinfache

425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 3 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 3 π}{4})) : 5 cos (\frac{- arctan ( \frac{24}{7} ) + 7 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 7 π}{4})

z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + π}{4}), z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + 3 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 3 π}{4}), z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + 5 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 5 π}{4}), z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + 7 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 7 π}{4})

Beliebte Beispiele

4 x^{3} - 9 x^{2} + 6 x = 0

solvefor x,x^3=7x

so l v e f or x, x^{3} = 7 x

4 x^{3} + 6 x^{2} - 4 x = 0

x^{4} - 7 x^{2} + 6 x = 0

x^{9} = 1