erweitern ((x^7+4)^7)/(128x^{14)}

Lösung

erweitern

\frac{( x ^{7} + 4 ) ^{7}}{128 x ^{14}}

\frac{x ^{35}}{128} + \frac{7 x ^{28}}{32} + \frac{21 x ^{21}}{8} + \frac{35 x ^{14}}{2} + 70 x^{7} + 168 + \frac{224}{x ^{7}} + \frac{128}{x ^{14}}

Schritte zur Lösung

\frac{( x ^{7} + 4 ) ^{7}}{128 x ^{14}}

(x^{7} + 4)^{7} = x^{49} + 28 x^{42} + 336 x^{35} + 2240 x^{28} + 8960 x^{21} + 21504 x^{14} + 28672 x^{7} + 16384

= \frac{x ^{49} + 28 x ^{42} + 336 x ^{35} + 2240 x ^{28} + 8960 x ^{21} + 21504 x ^{14} + 28672 x ^{7} + 16384}{128 x ^{14}}

Wende Bruchregel an:

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

\frac{x ^{49} + 28 x ^{42} + 336 x ^{35} + 2240 x ^{28} + 8960 x ^{21} + 21504 x ^{14} + 28672 x ^{7} + 16384}{128 x ^{14}} = \frac{x ^{49}}{128 x ^{14}} + \frac{28 x ^{42}}{128 x ^{14}} + \frac{336 x ^{35}}{128 x ^{14}} + \frac{2240 x ^{28}}{128 x ^{14}} + \frac{8960 x ^{21}}{128 x ^{14}} + \frac{21504 x ^{14}}{128 x ^{14}} + \frac{28672 x ^{7}}{128 x ^{14}} + \frac{16384}{128 x ^{14}}

= \frac{x ^{49}}{128 x ^{14}} + \frac{28 x ^{42}}{128 x ^{14}} + \frac{336 x ^{35}}{128 x ^{14}} + \frac{2240 x ^{28}}{128 x ^{14}} + \frac{8960 x ^{21}}{128 x ^{14}} + \frac{21504 x ^{14}}{128 x ^{14}} + \frac{28672 x ^{7}}{128 x ^{14}} + \frac{16384}{128 x ^{14}}

\frac{x ^{49}}{128 x ^{14}} = \frac{x ^{35}}{128}

\frac{28 x ^{42}}{128 x ^{14}} = \frac{7 x ^{28}}{32}

\frac{336 x ^{35}}{128 x ^{14}} = \frac{21 x ^{21}}{8}

\frac{2240 x ^{28}}{128 x ^{14}} = \frac{35 x ^{14}}{2}

\frac{8960 x ^{21}}{128 x ^{14}} = 70 x^{7}

\frac{21504 x ^{14}}{128 x ^{14}} = 168

\frac{28672 x ^{7}}{128 x ^{14}} = \frac{224}{x ^{7}}

\frac{16384}{128 x ^{14}} = \frac{128}{x ^{14}}

= \frac{x ^{35}}{128} + \frac{7 x ^{28}}{32} + \frac{21 x ^{21}}{8} + \frac{35 x ^{14}}{2} + 70 x^{7} + 168 + \frac{224}{x ^{7}} + \frac{128}{x ^{14}}

e x p an d \frac{1}{( s ^{2} + 2 ) ^{3}}

e x p an d \frac{( 3 x - 1 ) ( 7 - 2 x )}{( 7 x ^{2} + 1 ) ( 10 x ^{3} - 1 )}

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

\frac{d}{d y} ((z) (x^{2} e^{y} - y z e^{x}))

e x p an d \frac{( 2 x + 1 ) ( 2 x - 3 )}{2}