erweitern (2+1/4 x)^8(2x-3/x)^2

Lösung

erweitern

(2 + \frac{1}{4} x)^{8} (2 x - \frac{3}{x})^{2}

\frac{256 ( 4 x ^{4} - 12 x ^{2} + 9 )}{x ^{2}} + \frac{256 ( 4 x ^{4} - 12 x ^{2} + 9 )}{x} + 448 x^{4} - 1344 x^{2} + 1008 + 112 x^{5} - 336 x^{3} + 252 x + \frac{35 ( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{2}}{8} + \frac{7 ( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{3}}{16} + \frac{7 ( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{4}}{256} + \frac{( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{5}}{1024} + \frac{( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{6}}{65536}

Schritte zur Lösung

(2 + \frac{1}{4} x)^{8} (2 x - \frac{3}{x})^{2}

Schreibe

(2 x - \frac{3}{x})^{2}

um:

\frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}}

= (2 + \frac{1}{4} x)^{8} \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}}

= \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} (2 + \frac{1}{4} x)^{8}

Schreibe

(2 + \frac{1}{4} x)^{8}

um:

256 + 256 x + 112 x^{2} + 28 x^{3} + \frac{35 x ^{4}}{8} + \frac{7 x ^{5}}{16} + \frac{7 x ^{6}}{256} + \frac{x ^{7}}{1024} + \frac{x ^{8}}{65536}

= \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} (256 + 256 x + 112 x^{2} + 28 x^{3} + \frac{35 x ^{4}}{8} + \frac{7 x ^{5}}{16} + \frac{7 x ^{6}}{256} + \frac{x ^{7}}{1024} + \frac{x ^{8}}{65536})

Setze Klammern

= \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot 256 + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot 256 x + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot 112 x^{2} + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot 28 x^{3} + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot \frac{35 x ^{4}}{8} + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot \frac{7 x ^{5}}{16} + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot \frac{7 x ^{6}}{256} + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot \frac{x ^{7}}{1024} + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot \frac{x ^{8}}{65536}

Vereinfache

\frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot 256 + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot 256 x + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot 112 x^{2} + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot 28 x^{3} + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot \frac{35 x ^{4}}{8} + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot \frac{7 x ^{5}}{16} + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot \frac{7 x ^{6}}{256} + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot \frac{x ^{7}}{1024} + \frac{4 x ^{4} - 12 x ^{2} + 9}{x ^{2}} \cdot \frac{x ^{8}}{65536} : \frac{256 ( 4 x ^{4} - 12 x ^{2} + 9 )}{x ^{2}} + \frac{256 ( 4 x ^{4} - 12 x ^{2} + 9 )}{x} + 448 x^{4} - 1344 x^{2} + 1008 + 112 x^{5} - 336 x^{3} + 252 x + \frac{35 ( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{2}}{8} + \frac{7 ( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{3}}{16} + \frac{7 ( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{4}}{256} + \frac{( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{5}}{1024} + \frac{( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{6}}{65536}

= \frac{256 ( 4 x ^{4} - 12 x ^{2} + 9 )}{x ^{2}} + \frac{256 ( 4 x ^{4} - 12 x ^{2} + 9 )}{x} + 448 x^{4} - 1344 x^{2} + 1008 + 112 x^{5} - 336 x^{3} + 252 x + \frac{35 ( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{2}}{8} + \frac{7 ( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{3}}{16} + \frac{7 ( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{4}}{256} + \frac{( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{5}}{1024} + \frac{( 4 x ^{4} - 12 x ^{2} + 9 ) x ^{6}}{65536}

\frac{\partial}{\partial x} ((z) (x^{2} yz - e^{z}))

e x p an d, \frac{585}{13} + (28 - 15) \cdot 5

e x p an d (a - 2) l e f t (a - 12)

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

e x p an d \frac{n ( n + 1 ) ( n + 2 )}{6}