de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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 = - \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}

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})) : - \frac{4 + 10 - 2 5}{2} + i \frac{4 - 10 - 2 5}{2}

Vereinfache

542 (cos (\frac{\frac{π}{4} + 2 \cdot 3 π}{5}) + i sin (\frac{\frac{π}{4} + 2 \cdot 3 π}{5})) : - 1 - i

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 = - \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}

Beliebte Beispiele

x^{3} = 36

3 x^{25} x - 2 = 0

x^{4} - 10 x^{2} + 16 = 0

2x^4-2^{1/2}x^3-32x^2+8*2^{1/2}x+128=0

2 x^{4} - 2^{\frac{1}{2}} x^{3} - 32 x^{2} + 8 \cdot 2^{\frac{1}{2}} x + 128 = 0

(x-6)^4(x-8)^4=272

(x - 6)^{4} (x - 8)^{4} = 272