7pik^3=22704

Lösung

7 π k^{3} = 22704

k = \frac{2 3 2838}{3 7 3 π}, k = - \frac{7 ^{\frac{2}{3}} 3 2838 π ^{\frac{2}{3}}}{7 π} + i \frac{3 ^{\frac{5}{6}} \cdot 7 ^{\frac{2}{3}} 3 946 π ^{\frac{2}{3}}}{7 π}, k = - \frac{7 ^{\frac{2}{3}} 3 2838 π ^{\frac{2}{3}}}{7 π} - i \frac{3 ^{\frac{5}{6}} \cdot 7 ^{\frac{2}{3}} 3 946 π ^{\frac{2}{3}}}{7 π}

Schritte zur Lösung

7 π k^{3} = 22704

Teile beide Seiten durch

7 π

k^{3} = \frac{22704}{7 π}

Für

x^{3} = f (a)

sind die Lösungen

x = 3 f (a), 3 f (a) \frac{- 1 - 3 i}{2}, 3 f (a) \frac{- 1 + 3 i}{2}

k = 3 \frac{22704}{7 π}, k = 3 \frac{22704}{7 π} \frac{- 1 + 3 i}{2}, k = 3 \frac{22704}{7 π} \frac{- 1 - 3 i}{2}

3 \frac{22704}{7 π} = \frac{2 3 2838}{3 7 3 π}

Vereinfache

3 \frac{22704}{7 π} \frac{- 1 + 3 i}{2} : - \frac{7 ^{\frac{2}{3}} 3 2838 π ^{\frac{2}{3}}}{7 π} + i \frac{3 ^{\frac{5}{6}} \cdot 7 ^{\frac{2}{3}} 3 946 π ^{\frac{2}{3}}}{7 π}

Vereinfache

3 \frac{22704}{7 π} \frac{- 1 - 3 i}{2} : - \frac{7 ^{\frac{2}{3}} 3 2838 π ^{\frac{2}{3}}}{7 π} - i \frac{3 ^{\frac{5}{6}} \cdot 7 ^{\frac{2}{3}} 3 946 π ^{\frac{2}{3}}}{7 π}

k = \frac{2 3 2838}{3 7 3 π}, k = - \frac{7 ^{\frac{2}{3}} 3 2838 π ^{\frac{2}{3}}}{7 π} + i \frac{3 ^{\frac{5}{6}} \cdot 7 ^{\frac{2}{3}} 3 946 π ^{\frac{2}{3}}}{7 π}, k = - \frac{7 ^{\frac{2}{3}} 3 2838 π ^{\frac{2}{3}}}{7 π} - i \frac{3 ^{\frac{5}{6}} \cdot 7 ^{\frac{2}{3}} 3 946 π ^{\frac{2}{3}}}{7 π}

x = x^{3} - 2 x^{2} + 1

x^{3} - 2 x^{2} + x - 1 = - x^{2} + 3 x - 1

30 x^{4} - 30 x^{2} = 0

4 x^{3} = x^{3} + 3 x^{2} + 2 x

3 x^{4} - 7 x - 3 x^{2} + 5 x + 2 = 0