solvefor V= 4/3 pir^3,r

Lösung

löse nach

V = \frac{4}{3} π r^{3}, r

r = 3 \frac{3 V}{4 π}, r = - \frac{4 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 V}{8 π} + i \frac{3 ^{\frac{5}{6}} \cdot 4 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 V}{8 π}, r = - \frac{4 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 V}{8 π} - i \frac{3 ^{\frac{5}{6}} \cdot 4 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 V}{8 π}

Schritte zur Lösung

V = \frac{4}{3} π r^{3}

Tausche die Seiten

\frac{4}{3} π r^{3} = V

Vereinfache

\frac{4}{3} π : \frac{4 π}{3}

\frac{4 π}{3} r^{3} = V

Multipliziere beide Seiten mit

3

4 π r^{3} = 3 V

Teile beide Seiten durch

4 π

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

Vereinfache

3 \frac{3 V}{4 π} \frac{- 1 + 3 i}{2} : - \frac{4 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 V}{8 π} + i \frac{3 ^{\frac{5}{6}} \cdot 4 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 V}{8 π}

Vereinfache

3 \frac{3 V}{4 π} \frac{- 1 - 3 i}{2} : - \frac{4 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 V}{8 π} - i \frac{3 ^{\frac{5}{6}} \cdot 4 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 V}{8 π}

r = 3 \frac{3 V}{4 π}, r = - \frac{4 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 V}{8 π} + i \frac{3 ^{\frac{5}{6}} \cdot 4 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 V}{8 π}, r = - \frac{4 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 V}{8 π} - i \frac{3 ^{\frac{5}{6}} \cdot 4 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 V}{8 π}

(x^{3} - 1) (x^{5} - 1) = 0

\frac{2 x ^{4} - 1}{8} = 0

y^{3} - 3 = 1

(x^{2} + 4)^{2} - 13 (x^{2} + 4) + 40 = 0

125 x^{4} + x = 0