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Beliebt Algebra >

z^4=1+i

Lösung

z^{4} = 1 + i

Lösung

z = \frac{8 2 2 + 2 + 2}{2} + i \frac{8 2 2 - 2 + 2}{2}, z = - \frac{8 2 2 - 2 + 2}{2} + i \frac{8 2 2 + 2 + 2}{2}, z = - \frac{8 2 2 + 2 + 2}{2} - i \frac{8 2 2 - 2 + 2}{2}, z = \frac{8 2 2 - 2 + 2}{2} - i \frac{8 2 2 + 2 + 2}{2}

Schritte zur Lösung

z^{4} = 1 + 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 = 1 + i

∣ a ∣ = 2

ar g (a) = \frac{π}{4}

z = 42 (cos (\frac{\frac{π}{4} + 2 \cdot 0 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 0 π}{4})), z = 42 (cos (\frac{\frac{π}{4} + 2 \cdot 1 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 1 π}{4})), z = 42 (cos (\frac{\frac{π}{4} + 2 \cdot 2 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 2 π}{4})), z = 42 (cos (\frac{\frac{π}{4} + 2 \cdot 3 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 3 π}{4}))

Vereinfache

42 (cos (\frac{\frac{π}{4} + 2 \cdot 0 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 0 π}{4})) : \frac{8 2 2 + 2 + 2}{2} + i \frac{8 2 2 - 2 + 2}{2}

Vereinfache

42 (cos (\frac{\frac{π}{4} + 2 \cdot 1 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 1 π}{4})) : - \frac{8 2 2 - 2 + 2}{2} + i \frac{8 2 2 + 2 + 2}{2}

Vereinfache

42 (cos (\frac{\frac{π}{4} + 2 \cdot 2 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 2 π}{4})) : - \frac{8 2 2 + 2 + 2}{2} - i \frac{8 2 2 - 2 + 2}{2}

Vereinfache

42 (cos (\frac{\frac{π}{4} + 2 \cdot 3 π}{4}) + i sin (\frac{\frac{π}{4} + 2 \cdot 3 π}{4})) : \frac{8 2 2 - 2 + 2}{2} - i \frac{8 2 2 + 2 + 2}{2}

z = \frac{8 2 2 + 2 + 2}{2} + i \frac{8 2 2 - 2 + 2}{2}, z = - \frac{8 2 2 - 2 + 2}{2} + i \frac{8 2 2 + 2 + 2}{2}, z = - \frac{8 2 2 + 2 + 2}{2} - i \frac{8 2 2 - 2 + 2}{2}, z = \frac{8 2 2 - 2 + 2}{2} - i \frac{8 2 2 + 2 + 2}{2}

Beliebte Beispiele

5 x^{3} - 6 x^{2} - 7 = 0

0 = x^{4} - 2 x^{2} - 8

solvefor y,x=y^6

so l v e f or y, x = y^{6}

solvefor y,x=y^7

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

z^{4} + 81 = 0