pi*r^3= 1/4

Lösung

π \cdot r^{3} = \frac{1}{4}

r = 3 \frac{1}{4 π}, r = - \frac{4 ^{\frac{2}{3}} π ^{\frac{2}{3}}}{8 π} + i \frac{4 ^{\frac{2}{3}} 3 π ^{\frac{2}{3}}}{8 π}, r = - \frac{4 ^{\frac{2}{3}} π ^{\frac{2}{3}}}{8 π} - i \frac{4 ^{\frac{2}{3}} 3 π ^{\frac{2}{3}}}{8 π}

Schritte zur Lösung

π r^{3} = \frac{1}{4}

Teile beide Seiten durch

π

r^{3} = \frac{1}{4 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}

r = 3 \frac{1}{4 π}, r = 3 \frac{1}{4 π} \frac{- 1 + 3 i}{2}, r = 3 \frac{1}{4 π} \frac{- 1 - 3 i}{2}

Vereinfache

3 \frac{1}{4 π} \frac{- 1 + 3 i}{2} : - \frac{4 ^{\frac{2}{3}} π ^{\frac{2}{3}}}{8 π} + i \frac{4 ^{\frac{2}{3}} 3 π ^{\frac{2}{3}}}{8 π}

Vereinfache

3 \frac{1}{4 π} \frac{- 1 - 3 i}{2} : - \frac{4 ^{\frac{2}{3}} π ^{\frac{2}{3}}}{8 π} - i \frac{4 ^{\frac{2}{3}} 3 π ^{\frac{2}{3}}}{8 π}

r = 3 \frac{1}{4 π}, r = - \frac{4 ^{\frac{2}{3}} π ^{\frac{2}{3}}}{8 π} + i \frac{4 ^{\frac{2}{3}} 3 π ^{\frac{2}{3}}}{8 π}, r = - \frac{4 ^{\frac{2}{3}} π ^{\frac{2}{3}}}{8 π} - i \frac{4 ^{\frac{2}{3}} 3 π ^{\frac{2}{3}}}{8 π}

c^{3} - 5 c^{2} + 7 c - 109 = 0

so l v e f or x, f = 5 x^{3} + 1 f (- 8)

- 12 = 5 (6 x^{4}) 7 (x^{6})

(x + 1)^{3} - 27 = 0

x^{3} + x^{2} = 6 x