erweitern ((x+1)/(2x))^3

Lösung

erweitern

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

\frac{1}{8} + \frac{3}{8 x} + \frac{3}{8 x ^{2}} + \frac{1}{8 x ^{3}}

Schritte zur Lösung

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

Wende Exponentenregel an:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Wende Exponentenregel an:

(a \cdot b)^{n} = a^{n} b^{n}

(2 x)^{3} = 2^{3} x^{3}

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

(x + 1)^{3} = x^{3} + 3 x^{2} + 3 x + 1

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

2^{3} = 8

= \frac{x ^{3} + 3 x ^{2} + 3 x + 1}{8 x ^{3}}

Wende Bruchregel an:

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

\frac{x ^{3} + 3 x ^{2} + 3 x + 1}{8 x ^{3}} = \frac{x ^{3}}{8 x ^{3}} + \frac{3 x ^{2}}{8 x ^{3}} + \frac{3 x}{8 x ^{3}} + \frac{1}{8 x ^{3}}

= \frac{x ^{3}}{8 x ^{3}} + \frac{3 x ^{2}}{8 x ^{3}} + \frac{3 x}{8 x ^{3}} + \frac{1}{8 x ^{3}}

Streiche

\frac{x ^{3}}{8 x ^{3}} : \frac{1}{8}

= \frac{1}{8} + \frac{3 x ^{2}}{8 x ^{3}} + \frac{3 x}{8 x ^{3}} + \frac{1}{8 x ^{3}}

Streiche

\frac{3 x ^{2}}{8 x ^{3}} : \frac{3}{8 x}

= \frac{1}{8} + \frac{3}{8 x} + \frac{3 x}{8 x ^{3}} + \frac{1}{8 x ^{3}}

Streiche

\frac{3 x}{8 x ^{3}} : \frac{3}{8 x ^{2}}

= \frac{1}{8} + \frac{3}{8 x} + \frac{3}{8 x ^{2}} + \frac{1}{8 x ^{3}}

e x p an d \frac{12 x + 7}{4} - \frac{30 - 18 x}{6}

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

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

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

e x p an d Z {y (n + 2)}