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Beliebt Algebra >

z^5=4-4i

Lösung

z^{5} = 4 - 4 i

Lösung

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

Schritte zur Lösung

z^{5} = 4 - 4 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 = 4 - 4 i

∣ a ∣ = 42

ar g (a) = - \frac{π}{4}

z = 542 (cos (\frac{- \frac{π}{4} + 2 \cdot 0 π}{5}) + i sin (\frac{- \frac{π}{4} + 2 \cdot 0 π}{5})), z = 542 (cos (\frac{- \frac{π}{4} + 2 \cdot 1 π}{5}) + i sin (\frac{- \frac{π}{4} + 2 \cdot 1 π}{5})), z = 542 (cos (\frac{- \frac{π}{4} + 2 \cdot 2 π}{5}) + i sin (\frac{- \frac{π}{4} + 2 \cdot 2 π}{5})), z = 542 (cos (\frac{- \frac{π}{4} + 2 \cdot 3 π}{5}) + i sin (\frac{- \frac{π}{4} + 2 \cdot 3 π}{5})), z = 542 (cos (\frac{- \frac{π}{4} + 2 \cdot 4 π}{5}) + i sin (\frac{- \frac{π}{4} + 2 \cdot 4 π}{5}))

Vereinfache

542 (cos (\frac{- \frac{π}{4} + 2 \cdot 0 π}{5}) + i sin (\frac{- \frac{π}{4} + 2 \cdot 0 π}{5})) : \frac{4 + 10 + 2 5}{2} - i \frac{4 - 10 + 2 5}{2}

Vereinfache

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

Vereinfache

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

Vereinfache

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

Vereinfache

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

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

Beliebte Beispiele

(- x)^{4} = - (x)^{4}

(x-8)^2(x-1)^2=x^2

(x - 8)^{2} (x - 1)^{2} = x^{2}

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

8x^3+x^2-32x-4=0

8 x^{3} + x^{2} - 32 x - 4 = 0

solvefor x,x^3-a^2x=0

so l v e f or x, x^{3} - a^{2} x = 0