erweitern ((x^2+y^2)^2)/(x^2y^2)

Lösung

erweitern

\frac{( x ^{2} + y ^{2} ) ^{2}}{x ^{2} y ^{2}}

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

Schritte zur Lösung

\frac{( x ^{2} + y ^{2} ) ^{2}}{x ^{2} y ^{2}}

(x^{2} + y^{2})^{2} = x^{4} + 2 x^{2} y^{2} + y^{4}

= \frac{x ^{4} + 2 x ^{2} y ^{2} + y ^{4}}{x ^{2} y ^{2}}

Wende Bruchregel an:

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

\frac{x ^{4} + 2 x ^{2} y ^{2} + y ^{4}}{x ^{2} y ^{2}} = \frac{x ^{4}}{x ^{2} y ^{2}} + \frac{2 x ^{2} y ^{2}}{x ^{2} y ^{2}} + \frac{y ^{4}}{x ^{2} y ^{2}}

= \frac{x ^{4}}{x ^{2} y ^{2}} + \frac{2 x ^{2} y ^{2}}{x ^{2} y ^{2}} + \frac{y ^{4}}{x ^{2} y ^{2}}

Streiche

\frac{x ^{4}}{x ^{2} y ^{2}} : \frac{x ^{2}}{y ^{2}}

= \frac{x ^{2}}{y ^{2}} + \frac{2 x ^{2} y ^{2}}{x ^{2} y ^{2}} + \frac{y ^{4}}{x ^{2} y ^{2}}

Streiche

\frac{2 x ^{2} y ^{2}}{x ^{2} y ^{2}} : 2

= \frac{x ^{2}}{y ^{2}} + 2 + \frac{y ^{4}}{x ^{2} y ^{2}}

Streiche

\frac{y ^{4}}{x ^{2} y ^{2}} : \frac{y ^{2}}{x ^{2}}

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

e x p an d \frac{1}{z ( z + 1 ) ^{2}}

e x p an d \frac{( 3 mn - 27 m ^{3 n} )}{( 1 + 3 m ) ( 1 - 3 m )}

e x p an d \frac{n ^{2} + n - 1}{( 2 n + 2 ) ^{2}}

\frac{d}{d x} (\frac{( 10 x ^{2} - 7 x - 12 ) x}{( 10 z + 8 ) ( 2 x + 3 )})

e x p an d 3242