erweitern ((x+y)(x+z)(y+z))/(xyz)

Lösung

erweitern

\frac{( x + y ) ( x + z ) ( y + z )}{x yz}

\frac{x}{z} + \frac{y}{z} + 2 + \frac{y}{x} + \frac{z}{x} + \frac{x}{y} + \frac{z}{y}

Schritte zur Lösung

\frac{( x + y ) ( x + z ) ( y + z )}{x yz}

Multipliziere aus

(x + y) (x + z) (y + z) : x^{2} y + x y^{2} + 2 x yz + y^{2} z + y z^{2} + x^{2} z + x z^{2}

= \frac{x ^{2} y + x y ^{2} + 2 x yz + y ^{2} z + y z ^{2} + x ^{2} z + x z ^{2}}{x yz}

Wende Bruchregel an:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{x ^{2} y + x y ^{2} + 2 x yz + y ^{2} z + y z ^{2} + x ^{2} z + x z ^{2}}{x yz} = \frac{x ^{2} y}{x yz} + \frac{x y ^{2}}{x yz} + \frac{2 x yz}{x yz} + \frac{y ^{2} z}{x yz} + \frac{y z ^{2}}{x yz} + \frac{x ^{2} z}{x yz} + \frac{x z ^{2}}{x yz}

= \frac{x ^{2} y}{x yz} + \frac{x y ^{2}}{x yz} + \frac{2 x yz}{x yz} + \frac{y ^{2} z}{x yz} + \frac{y z ^{2}}{x yz} + \frac{x ^{2} z}{x yz} + \frac{x z ^{2}}{x yz}

\frac{x ^{2} y}{x yz} = \frac{x}{z}

\frac{x y ^{2}}{x yz} = \frac{y}{z}

\frac{2 x yz}{x yz} = 2

\frac{y ^{2} z}{x yz} = \frac{y}{x}

\frac{y z ^{2}}{x yz} = \frac{z}{x}

\frac{x ^{2} z}{x yz} = \frac{x}{y}

\frac{x z ^{2}}{x yz} = \frac{z}{y}

= \frac{x}{z} + \frac{y}{z} + 2 + \frac{y}{x} + \frac{z}{x} + \frac{x}{y} + \frac{z}{y}

e x p an d P (- 5, 2) Q (2, - 1)

e x p an d \frac{e ^{- 3 s}}{s ( s ^{2} + 9 )}

e x p an d (5 + 2 i)^{3} + (5 - 2 i)^{3}

e x p an d 3 (2 + 7)

e x p an d (x y + y^{2}) d x + (x + 2 y - 1) d y