erweitern (t^{1/2}x^{1/2}+t^{1/3}y^{1/3})^3

Lösung

erweitern

(t^{\frac{1}{2}} x^{\frac{1}{2}} + t^{\frac{1}{3}} y^{\frac{1}{3}})^{3}

t^{\frac{3}{2}} x^{\frac{3}{2}} + 3 t^{\frac{4}{3}} x y^{\frac{1}{3}} + 3 t^{\frac{7}{6}} x^{\frac{1}{2}} y^{\frac{2}{3}} + t y

Schritte zur Lösung

(t^{\frac{1}{2}} x^{\frac{1}{2}} + t^{\frac{1}{3}} y^{\frac{1}{3}})^{3}

Wende Formel für perfekte dritte Potenzen an:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

(t^{\frac{1}{2}} x^{\frac{1}{2}} + t^{\frac{1}{3}} y^{\frac{1}{3}})^{3} = (t^{\frac{1}{2}} x^{\frac{1}{2}})^{3} + 3 (t^{\frac{1}{2}} x^{\frac{1}{2}})^{2} t^{\frac{1}{3}} y^{\frac{1}{3}} + 3 t^{\frac{1}{2}} x^{\frac{1}{2}} (t^{\frac{1}{3}} y^{\frac{1}{3}})^{2} + (t^{\frac{1}{3}} y^{\frac{1}{3}})^{3}

= (t^{\frac{1}{2}} x^{\frac{1}{2}})^{3} + 3 (t^{\frac{1}{2}} x^{\frac{1}{2}})^{2} t^{\frac{1}{3}} y^{\frac{1}{3}} + 3 t^{\frac{1}{2}} x^{\frac{1}{2}} (t^{\frac{1}{3}} y^{\frac{1}{3}})^{2} + (t^{\frac{1}{3}} y^{\frac{1}{3}})^{3}

(t^{\frac{1}{2}} x^{\frac{1}{2}})^{3} = t^{\frac{3}{2}} x^{\frac{3}{2}}

3 (t^{\frac{1}{2}} x^{\frac{1}{2}})^{2} t^{\frac{1}{3}} y^{\frac{1}{3}} = 3 t^{\frac{4}{3}} x y^{\frac{1}{3}}

3 t^{\frac{1}{2}} x^{\frac{1}{2}} (t^{\frac{1}{3}} y^{\frac{1}{3}})^{2} = 3 t^{\frac{7}{6}} x^{\frac{1}{2}} y^{\frac{2}{3}}

(t^{\frac{1}{3}} y^{\frac{1}{3}})^{3} = t y

= t^{\frac{3}{2}} x^{\frac{3}{2}} + 3 t^{\frac{4}{3}} x y^{\frac{1}{3}} + 3 t^{\frac{7}{6}} x^{\frac{1}{2}} y^{\frac{2}{3}} + t y

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

e x p an d (3 a + 3 b)^{3}

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

e x p an d - \frac{6}{( x + 2 ) ^{2}}

e x p an d \frac{1}{( j w ) ( a + j w )}