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Beliebt Algebra >

erweitern (((x+1))/(2x))^9

Lösung

erweitern

(\frac{( x + 1 )}{2 x})^{9}

Lösung

\frac{1}{512} + \frac{9}{512 x} + \frac{9}{128 x ^{2}} + \frac{21}{128 x ^{3}} + \frac{63}{256 x ^{4}} + \frac{63}{256 x ^{5}} + \frac{21}{128 x ^{6}} + \frac{9}{128 x ^{7}} + \frac{9}{512 x ^{8}} + \frac{1}{512 x ^{9}}

Schritte zur Lösung

(\frac{x + 1}{2 x})^{9}

Wende Exponentenregel an:

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

= \frac{( x + 1 ) ^{9}}{( 2 x ) ^{9}}

Wende Exponentenregel an:

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

(2 x)^{9} = 2^{9} x^{9}

= \frac{( x + 1 ) ^{9}}{2 ^{9} x ^{9}}

(x + 1)^{9} = x^{9} + 9 x^{8} + 36 x^{7} + 84 x^{6} + 126 x^{5} + 126 x^{4} + 84 x^{3} + 36 x^{2} + 9 x + 1

= \frac{x ^{9} + 9 x ^{8} + 36 x ^{7} + 84 x ^{6} + 126 x ^{5} + 126 x ^{4} + 84 x ^{3} + 36 x ^{2} + 9 x + 1}{2 ^{9} x ^{9}}

2^{9} = 512

= \frac{x ^{9} + 9 x ^{8} + 36 x ^{7} + 84 x ^{6} + 126 x ^{5} + 126 x ^{4} + 84 x ^{3} + 36 x ^{2} + 9 x + 1}{512 x ^{9}}

Wende Bruchregel an:

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

\frac{x ^{9} + 9 x ^{8} + 36 x ^{7} + 84 x ^{6} + 126 x ^{5} + 126 x ^{4} + 84 x ^{3} + 36 x ^{2} + 9 x + 1}{512 x ^{9}} = \frac{x ^{9}}{512 x ^{9}} + \frac{9 x ^{8}}{512 x ^{9}} + \frac{36 x ^{7}}{512 x ^{9}} + \frac{84 x ^{6}}{512 x ^{9}} + \frac{126 x ^{5}}{512 x ^{9}} + \frac{126 x ^{4}}{512 x ^{9}} + \frac{84 x ^{3}}{512 x ^{9}} + \frac{36 x ^{2}}{512 x ^{9}} + \frac{9 x}{512 x ^{9}} + \frac{1}{512 x ^{9}}

= \frac{x ^{9}}{512 x ^{9}} + \frac{9 x ^{8}}{512 x ^{9}} + \frac{36 x ^{7}}{512 x ^{9}} + \frac{84 x ^{6}}{512 x ^{9}} + \frac{126 x ^{5}}{512 x ^{9}} + \frac{126 x ^{4}}{512 x ^{9}} + \frac{84 x ^{3}}{512 x ^{9}} + \frac{36 x ^{2}}{512 x ^{9}} + \frac{9 x}{512 x ^{9}} + \frac{1}{512 x ^{9}}

\frac{x ^{9}}{512 x ^{9}} = \frac{1}{512}

\frac{9 x ^{8}}{512 x ^{9}} = \frac{9}{512 x}

\frac{36 x ^{7}}{512 x ^{9}} = \frac{9}{128 x ^{2}}

\frac{84 x ^{6}}{512 x ^{9}} = \frac{21}{128 x ^{3}}

\frac{126 x ^{5}}{512 x ^{9}} = \frac{63}{256 x ^{4}}

\frac{126 x ^{4}}{512 x ^{9}} = \frac{63}{256 x ^{5}}

\frac{84 x ^{3}}{512 x ^{9}} = \frac{21}{128 x ^{6}}

\frac{36 x ^{2}}{512 x ^{9}} = \frac{9}{128 x ^{7}}

\frac{9 x}{512 x ^{9}} = \frac{9}{512 x ^{8}}

= \frac{1}{512} + \frac{9}{512 x} + \frac{9}{128 x ^{2}} + \frac{21}{128 x ^{3}} + \frac{63}{256 x ^{4}} + \frac{63}{256 x ^{5}} + \frac{21}{128 x ^{6}} + \frac{9}{128 x ^{7}} + \frac{9}{512 x ^{8}} + \frac{1}{512 x ^{9}}

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