erweitern ((k^{14}+4k^7+3))/(2k^{11)+2k^4}

Lösung

erweitern

\frac{( k ^{14} + 4 k ^{7} + 3 )}{2 k ^{11} + 2 k ^{4}}

\frac{k ^{3}}{2} - \frac{3 ^{\frac{3}{7}}}{2} + \frac{7 3 \cdot 3 ^{\frac{2}{7}}}{2} + \frac{3 ^{\frac{4}{7}}}{2 k} - \frac{7 3 \cdot 3 ^{\frac{3}{7}}}{2 k} - \frac{3 ^{\frac{5}{7}}}{2 k ^{2}} + \frac{7 3 \cdot 3 ^{\frac{4}{7}}}{2 k ^{2}} + \frac{3 ^{\frac{6}{7}}}{2 k ^{3}} - \frac{7 3 \cdot 3 ^{\frac{5}{7}}}{2 k ^{3}} + \frac{7 3 \cdot 3 ^{\frac{6}{7}}}{2 k ^{4}}

Schritte zur Lösung

\frac{k ^{14} + 4 k ^{7} + 3}{2 k ^{11} + 2 k ^{4}}

Faktorisiere

k^{14} + 4 k^{7} + 3 : (k + 1) (k + 73) (k^{6} - k^{5} + k^{4} - k^{3} + k^{2} - k + 1) (k^{6} - 73 k^{5} + 3^{\frac{2}{7}} k^{4} - 3^{\frac{3}{7}} k^{3} + 3^{\frac{4}{7}} k^{2} - 3^{\frac{5}{7}} k + 3^{\frac{6}{7}})

= \frac{( k + 1 ) ( k + 7 3 ) ( k ^{6} - k ^{5} + k ^{4} - k ^{3} + k ^{2} - k + 1 ) ( k ^{6} - 7 3 k ^{5} + 3 ^{\frac{2}{7}} k ^{4} - 3 ^{\frac{3}{7}} k ^{3} + 3 ^{\frac{4}{7}} k ^{2} - 3 ^{\frac{5}{7}} k + 3 ^{\frac{6}{7}} )}{2 k ^{11} + 2 k ^{4}}

Faktorisiere

2 k^{11} + 2 k^{4} : 2 k^{4} (k + 1) (k^{6} - k^{5} + k^{4} - k^{3} + k^{2} - k + 1)

= \frac{( k + 1 ) ( k + 7 3 ) ( k ^{6} - k ^{5} + k ^{4} - k ^{3} + k ^{2} - k + 1 ) ( k ^{6} - 7 3 k ^{5} + 3 ^{\frac{2}{7}} k ^{4} - 3 ^{\frac{3}{7}} k ^{3} + 3 ^{\frac{4}{7}} k ^{2} - 3 ^{\frac{5}{7}} k + 3 ^{\frac{6}{7}} )}{2 k ^{4} ( k + 1 ) ( k ^{6} - k ^{5} + k ^{4} - k ^{3} + k ^{2} - k + 1 )}

Streiche

\frac{( k + 1 ) ( k + 7 3 ) ( k ^{6} - k ^{5} + k ^{4} - k ^{3} + k ^{2} - k + 1 ) ( k ^{6} - 7 3 k ^{5} + 3 ^{\frac{2}{7}} k ^{4} - 3 ^{\frac{3}{7}} k ^{3} + 3 ^{\frac{4}{7}} k ^{2} - 3 ^{\frac{5}{7}} k + 3 ^{\frac{6}{7}} )}{2 k ^{4} ( k + 1 ) ( k ^{6} - k ^{5} + k ^{4} - k ^{3} + k ^{2} - k + 1 )} : \frac{( k + 7 3 ) ( k ^{6} - 7 3 k ^{5} + 3 ^{\frac{2}{7}} k ^{4} - 3 ^{\frac{3}{7}} k ^{3} + 3 ^{\frac{4}{7}} k ^{2} - 3 ^{\frac{5}{7}} k + 3 ^{\frac{6}{7}} )}{2 k ^{4}}

= \frac{( k + 7 3 ) ( k ^{6} - 7 3 k ^{5} + 3 ^{\frac{2}{7}} k ^{4} - 3 ^{\frac{3}{7}} k ^{3} + 3 ^{\frac{4}{7}} k ^{2} - 3 ^{\frac{5}{7}} k + 3 ^{\frac{6}{7}} )}{2 k ^{4}}

Multipliziere aus

(k + 73) (k^{6} - 73 k^{5} + 3^{\frac{2}{7}} k^{4} - 3^{\frac{3}{7}} k^{3} + 3^{\frac{4}{7}} k^{2} - 3^{\frac{5}{7}} k + 3^{\frac{6}{7}}) : k^{7} - 3^{\frac{3}{7}} k^{4} + 73 \cdot 3^{\frac{2}{7}} k^{4} + 3^{\frac{4}{7}} k^{3} - 73 \cdot 3^{\frac{3}{7}} k^{3} - 3^{\frac{5}{7}} k^{2} + 73 \cdot 3^{\frac{4}{7}} k^{2} + 3^{\frac{6}{7}} k - 73 \cdot 3^{\frac{5}{7}} k + 73 \cdot 3^{\frac{6}{7}}

= \frac{k ^{7} - 3 ^{\frac{3}{7}} k ^{4} + 7 3 \cdot 3 ^{\frac{2}{7}} k ^{4} + 3 ^{\frac{4}{7}} k ^{3} - 7 3 \cdot 3 ^{\frac{3}{7}} k ^{3} - 3 ^{\frac{5}{7}} k ^{2} + 7 3 \cdot 3 ^{\frac{4}{7}} k ^{2} + 3 ^{\frac{6}{7}} k - 7 3 \cdot 3 ^{\frac{5}{7}} k + 7 3 \cdot 3 ^{\frac{6}{7}}}{2 k ^{4}}

Wende Bruchregel an:

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

\frac{k ^{7} - 3 ^{\frac{3}{7}} k ^{4} + 7 3 \cdot 3 ^{\frac{2}{7}} k ^{4} + 3 ^{\frac{4}{7}} k ^{3} - 7 3 \cdot 3 ^{\frac{3}{7}} k ^{3} - 3 ^{\frac{5}{7}} k ^{2} + 7 3 \cdot 3 ^{\frac{4}{7}} k ^{2} + 3 ^{\frac{6}{7}} k - 7 3 \cdot 3 ^{\frac{5}{7}} k + 7 3 \cdot 3 ^{\frac{6}{7}}}{2 k ^{4}} = \frac{k ^{7}}{2 k ^{4}} - \frac{3 ^{\frac{3}{7}} k ^{4}}{2 k ^{4}} + \frac{7 3 \cdot 3 ^{\frac{2}{7}} k ^{4}}{2 k ^{4}} + \frac{3 ^{\frac{4}{7}} k ^{3}}{2 k ^{4}} - \frac{7 3 \cdot 3 ^{\frac{3}{7}} k ^{3}}{2 k ^{4}} - \frac{3 ^{\frac{5}{7}} k ^{2}}{2 k ^{4}} + \frac{7 3 \cdot 3 ^{\frac{4}{7}} k ^{2}}{2 k ^{4}} + \frac{3 ^{\frac{6}{7}} k}{2 k ^{4}} - \frac{7 3 \cdot 3 ^{\frac{5}{7}} k}{2 k ^{4}} + \frac{7 3 \cdot 3 ^{\frac{6}{7}}}{2 k ^{4}}

= \frac{k ^{7}}{2 k ^{4}} - \frac{3 ^{\frac{3}{7}} k ^{4}}{2 k ^{4}} + \frac{7 3 \cdot 3 ^{\frac{2}{7}} k ^{4}}{2 k ^{4}} + \frac{3 ^{\frac{4}{7}} k ^{3}}{2 k ^{4}} - \frac{7 3 \cdot 3 ^{\frac{3}{7}} k ^{3}}{2 k ^{4}} - \frac{3 ^{\frac{5}{7}} k ^{2}}{2 k ^{4}} + \frac{7 3 \cdot 3 ^{\frac{4}{7}} k ^{2}}{2 k ^{4}} + \frac{3 ^{\frac{6}{7}} k}{2 k ^{4}} - \frac{7 3 \cdot 3 ^{\frac{5}{7}} k}{2 k ^{4}} + \frac{7 3 \cdot 3 ^{\frac{6}{7}}}{2 k ^{4}}

\frac{k ^{7}}{2 k ^{4}} = \frac{k ^{3}}{2}

\frac{3 ^{\frac{3}{7}} k ^{4}}{2 k ^{4}} = \frac{3 ^{\frac{3}{7}}}{2}

\frac{7 3 \cdot 3 ^{\frac{2}{7}} k ^{4}}{2 k ^{4}} = \frac{7 3 \cdot 3 ^{\frac{2}{7}}}{2}

\frac{3 ^{\frac{4}{7}} k ^{3}}{2 k ^{4}} = \frac{3 ^{\frac{4}{7}}}{2 k}

\frac{7 3 \cdot 3 ^{\frac{3}{7}} k ^{3}}{2 k ^{4}} = \frac{7 3 \cdot 3 ^{\frac{3}{7}}}{2 k}

\frac{3 ^{\frac{5}{7}} k ^{2}}{2 k ^{4}} = \frac{3 ^{\frac{5}{7}}}{2 k ^{2}}

\frac{7 3 \cdot 3 ^{\frac{4}{7}} k ^{2}}{2 k ^{4}} = \frac{7 3 \cdot 3 ^{\frac{4}{7}}}{2 k ^{2}}

\frac{3 ^{\frac{6}{7}} k}{2 k ^{4}} = \frac{3 ^{\frac{6}{7}}}{2 k ^{3}}

\frac{7 3 \cdot 3 ^{\frac{5}{7}} k}{2 k ^{4}} = \frac{7 3 \cdot 3 ^{\frac{5}{7}}}{2 k ^{3}}

= \frac{k ^{3}}{2} - \frac{3 ^{\frac{3}{7}}}{2} + \frac{7 3 \cdot 3 ^{\frac{2}{7}}}{2} + \frac{3 ^{\frac{4}{7}}}{2 k} - \frac{7 3 \cdot 3 ^{\frac{3}{7}}}{2 k} - \frac{3 ^{\frac{5}{7}}}{2 k ^{2}} + \frac{7 3 \cdot 3 ^{\frac{4}{7}}}{2 k ^{2}} + \frac{3 ^{\frac{6}{7}}}{2 k ^{3}} - \frac{7 3 \cdot 3 ^{\frac{5}{7}}}{2 k ^{3}} + \frac{7 3 \cdot 3 ^{\frac{6}{7}}}{2 k ^{4}}

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

\frac{\partial}{\partial x} ((z) (z^{3} - z - x y))

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

\frac{d}{d x} \frac{( φ ( x ^{2} + 2 y ( φ , x ) ) + ψ ( φ , x , y ( φ , x ) ) ( x - 2 y ( φ , x ) ^{2} ) )}{x}

e x p an d \frac{e ^{- 2 π s}}{( s ^{2} + 1 ) \cdot ( s ^{2} - s + 4 )}