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Beliebt Algebra >

erweitern ((1+x)/2)^5

Lösung

erweitern

(\frac{1 + x}{2})^{5}

Lösung

\frac{x ^{5}}{32} + \frac{5 x ^{4}}{32} + \frac{5 x ^{3}}{16} + \frac{5 x ^{2}}{16} + \frac{5 x}{32} + \frac{1}{32}

Schritte zur Lösung

(\frac{1 + x}{2})^{5}

Wende Exponentenregel an:

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

= \frac{( 1 + x ) ^{5}}{2 ^{5}}

(1 + x)^{5} = 1 + 5 x + 10 x^{2} + 10 x^{3} + 5 x^{4} + x^{5}

= \frac{1 + 5 x + 10 x ^{2} + 10 x ^{3} + 5 x ^{4} + x ^{5}}{2 ^{5}}

2^{5} = 32

= \frac{1 + 5 x + 10 x ^{2} + 10 x ^{3} + 5 x ^{4} + x ^{5}}{32}

Wende Bruchregel an:

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

\frac{1 + 5 x + 10 x ^{2} + 10 x ^{3} + 5 x ^{4} + x ^{5}}{32} = \frac{1}{32} + \frac{5 x}{32} + \frac{10 x ^{2}}{32} + \frac{10 x ^{3}}{32} + \frac{5 x ^{4}}{32} + \frac{x ^{5}}{32}

= \frac{1}{32} + \frac{5 x}{32} + \frac{10 x ^{2}}{32} + \frac{10 x ^{3}}{32} + \frac{5 x ^{4}}{32} + \frac{x ^{5}}{32}

Streiche

\frac{10 x ^{2}}{32} : \frac{5 x ^{2}}{16}

= \frac{1}{32} + \frac{5 x}{32} + \frac{5 x ^{2}}{16} + \frac{10 x ^{3}}{32} + \frac{5 x ^{4}}{32} + \frac{x ^{5}}{32}

Streiche

\frac{10 x ^{3}}{32} : \frac{5 x ^{3}}{16}

= \frac{1}{32} + \frac{5 x}{32} + \frac{5 x ^{2}}{16} + \frac{5 x ^{3}}{16} + \frac{5 x ^{4}}{32} + \frac{x ^{5}}{32}

Rewrite in standard form

= \frac{x ^{5}}{32} + \frac{5 x ^{4}}{32} + \frac{5 x ^{3}}{16} + \frac{5 x ^{2}}{16} + \frac{5 x}{32} + \frac{1}{32}

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