erweitern x^{3/2}+x^{-1/3}

Lösung

erweitern

x^{\frac{3}{2}} + x^{- \frac{1}{3}}

\frac{1}{x ^{\frac{1}{3}}} + x^{\frac{3}{2}}

Schritte zur Lösung

x^{\frac{3}{2}} + x^{- \frac{1}{3}}

Wende Exponentenregel an:

a^{- b} = \frac{1}{a ^{b}}

= x^{\frac{3}{2}} + \frac{1}{x ^{\frac{1}{3}}}

Wandle das Element in einen Bruch um:

x^{\frac{3}{2}} = \frac{x ^{\frac{3}{2}} x ^{\frac{1}{3}}}{x ^{\frac{1}{3}}}

= \frac{1}{x ^{\frac{1}{3}}} + \frac{x ^{\frac{3}{2}} x ^{\frac{1}{3}}}{x ^{\frac{1}{3}}}

Da die Nenner gleich sind, fasse die Brüche zusammen.:

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

= \frac{1 + x ^{\frac{3}{2}} x ^{\frac{1}{3}}}{x ^{\frac{1}{3}}}

1 + x^{\frac{3}{2}} x^{\frac{1}{3}} = 1 + x^{\frac{3}{2} + \frac{1}{3}}

= \frac{1 + x ^{\frac{3}{2} + \frac{1}{3}}}{x ^{\frac{1}{3}}}

Füge

\frac{3}{2} + \frac{1}{3}

zusammen:

\frac{11}{6}

= \frac{1 + x ^{\frac{11}{6}}}{x ^{\frac{1}{3}}}

Wende Bruchregel an:

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

\frac{1 + x ^{\frac{11}{6}}}{x ^{\frac{1}{3}}} = \frac{1}{x ^{\frac{1}{3}}} + \frac{x ^{\frac{11}{6}}}{x ^{\frac{1}{3}}}

= \frac{1}{x ^{\frac{1}{3}}} + \frac{x ^{\frac{11}{6}}}{x ^{\frac{1}{3}}}

Streiche

\frac{x ^{\frac{11}{6}}}{x ^{\frac{1}{3}}} : x^{\frac{3}{2}}

= \frac{1}{x ^{\frac{1}{3}}} + x^{\frac{3}{2}}

e x p an d \frac{3 - 5 ( x + h ) ^{2} - 3 - 5 x ^{2}}{4 h}

e x p an d \frac{2 z ( z ^{2} + 2 z - 8 )}{( z - 1 ) ( z + 2 )}

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

e x p an d (8 x^{2} + 4 y) x^{2} + y^{2} = 1

e x p an d \frac{h ( α - ρ )}{α}