solvefor r,x= 5/3 pir^3

Lösung

löse nach

r, x = \frac{5}{3} π r^{3}

r = 3 \frac{3 x}{5 π}, r = - \frac{5 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{10 π} + i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 x}{10 π}, r = - \frac{5 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{10 π} - i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 x}{10 π}

Schritte zur Lösung

x = \frac{5}{3} π r^{3}

Tausche die Seiten

\frac{5}{3} π r^{3} = x

Vereinfache

\frac{5}{3} π : \frac{5 π}{3}

\frac{5 π}{3} r^{3} = x

Multipliziere beide Seiten mit

3

5 π r^{3} = 3 x

Teile beide Seiten durch

5 π

r^{3} = \frac{3 x}{5 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 x}{5 π}, r = 3 \frac{3 x}{5 π} \frac{- 1 + 3 i}{2}, r = 3 \frac{3 x}{5 π} \frac{- 1 - 3 i}{2}

Vereinfache

3 \frac{3 x}{5 π} \frac{- 1 + 3 i}{2} : - \frac{5 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{10 π} + i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 x}{10 π}

Vereinfache

3 \frac{3 x}{5 π} \frac{- 1 - 3 i}{2} : - \frac{5 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{10 π} - i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 x}{10 π}

r = 3 \frac{3 x}{5 π}, r = - \frac{5 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{10 π} + i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 x}{10 π}, r = - \frac{5 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{10 π} - i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 x}{10 π}

4 x^{3} - 6 x^{2} - 3 = \frac{1}{2}

x^{4} - 4 x^{2} - 45 = 0

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

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

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