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Beliebt Algebra >

faktorisieren 125x^9y^{18}-512z^{27}

Lösung

faktorisieren

125 x^{9} y^{18} - 512 z^{27}

Lösung

(5 x^{3} y^{6} - 8 z^{9}) (25 x^{6} y^{12} + 40 x^{3} y^{6} z^{9} + 64 z^{18})

Schritte zur Lösung

125 x^{9} y^{18} - 512 z^{27}

Schreibe

125 x^{9} y^{18}

um:

(5 x^{3} y^{6})^{3}

Schreibe

512 z^{27}

um:

(2 z^{3})^{9}

= (5 x^{3} y^{6})^{3} - (2 z^{3})^{9}

Wende Exponentenregel an:

a^{b c} = (a^{b})^{c}

(2 z^{3})^{9} = ((2 z^{3})^{3})^{3}

= (5 x^{3} y^{6})^{3} - ((2 z^{3})^{3})^{3}

Wende Formel zur Differenz von dritten Potenzen an:

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

(5 x^{3} y^{6})^{3} - ((2 z^{3})^{3})^{3} = (5 x^{3} y^{6} - (2 z^{3})^{3}) ((5 x^{3} y^{6})^{2} + (2 z^{3})^{3} 5 x^{3} y^{6} + (2 z^{3})^{6})

= (5 x^{3} y^{6} - (2 z^{3})^{3}) ((2 z^{3})^{6} + (2 z^{3})^{3} 5 x^{3} y^{6} + (5 x^{3} y^{6})^{2})

Vereinfache

(5 x^{3} y^{6} - (2 z^{3})^{3}) ((5 x^{3} y^{6})^{2} + (2 z^{3})^{3} \cdot 5 x^{3} y^{6} + (2 z^{3})^{6}) : (5 x^{3} y^{6} - 8 z^{9}) (25 x^{6} y^{12} + 40 x^{3} y^{6} z^{9} + 64 z^{18})

= (5 x^{3} y^{6} - 8 z^{9}) (25 x^{6} y^{12} + 40 x^{3} y^{6} z^{9} + 64 z^{18})

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