z^3=2^{5/2}(1+i)

Lösung

z^{3} = 2^{\frac{5}{2}} (1 + i)

z = \frac{1 + 3 ^{\frac{1}{2}}}{2 ^{\frac{1}{2}}} + i \frac{- 1 + 3}{2 ^{\frac{1}{2}}}, z = - 2^{\frac{1}{2}} + 2^{\frac{1}{2}} i, z = \frac{1 - 3 ^{\frac{1}{2}}}{2 ^{\frac{1}{2}}} + i \frac{- 1 - 3}{2 ^{\frac{1}{2}}}

Schritte zur Lösung

z^{3} = 2^{\frac{5}{2}} (1 + i)

Vereinfache

2^{\frac{5}{2}} (1 + i) : 4 \cdot 2^{\frac{1}{2}} + 4 \cdot 2^{\frac{1}{2}} 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 = 3, a = 4 \cdot 2^{\frac{1}{2}} + 4 \cdot 2^{\frac{1}{2}} i

∣ a ∣ = 8

ar g (a) = \frac{π}{4}

z = 38 (cos (\frac{\frac{π}{4} + 2 \cdot 0 π}{3}) + i sin (\frac{\frac{π}{4} + 2 \cdot 0 π}{3})), z = 38 (cos (\frac{\frac{π}{4} + 2 \cdot 1 π}{3}) + i sin (\frac{\frac{π}{4} + 2 \cdot 1 π}{3})), z = 38 (cos (\frac{\frac{π}{4} + 2 \cdot 2 π}{3}) + i sin (\frac{\frac{π}{4} + 2 \cdot 2 π}{3}))

Vereinfache

38 (cos (\frac{\frac{π}{4} + 2 \cdot 0 π}{3}) + i sin (\frac{\frac{π}{4} + 2 \cdot 0 π}{3})) : \frac{1 + 3 ^{\frac{1}{2}}}{2 ^{\frac{1}{2}}} + i \frac{- 1 + 3}{2 ^{\frac{1}{2}}}

Vereinfache

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

Vereinfache

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

z = \frac{1 + 3 ^{\frac{1}{2}}}{2 ^{\frac{1}{2}}} + i \frac{- 1 + 3}{2 ^{\frac{1}{2}}}, z = - 2^{\frac{1}{2}} + 2^{\frac{1}{2}} i, z = \frac{1 - 3 ^{\frac{1}{2}}}{2 ^{\frac{1}{2}}} + i \frac{- 1 - 3}{2 ^{\frac{1}{2}}}

4 x (x^{2} - 4) = 0

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

(1 - x) (1 - x) (1 - x) - (1 - x) = 0

x^{4} = \frac{1}{2}

30 x^{2} - 27 x - 3 x^{3} = 0