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Beliebt Algebra >

erweitern (x-1/(\sqrt[3]{x)})^{12}

Lösung

erweitern

(x - \frac{1}{3 x})^{12}

Lösung

x^{12} - 12 3 x^{32} + \frac{66 x ^{10} - 792 x ^{6} + 495 x ^{2}}{( 3 x ) ^{2}} - 220 x^{8} + 495 x^{6} 3 x^{2} - 792 x^{2} 3 x^{2} + 924 x^{4} - 220 + \frac{66}{3 x ^{4}} - \frac{12}{3 x ^{8}} + \frac{1}{x ^{4}}

Schritte zur Lösung

(x - \frac{1}{3 x})^{12}

Wende Binomialsetz an:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = x, b = - \frac{1}{3 x}

= i = 0 \sum 12 (i 12) x^{(12 - i)} (- \frac{1}{3 x})^{i}

Erweitere Summation

= \frac{12 !}{0 ! ( 12 - 0 ) !} x^{12} (- \frac{1}{3 x})^{0} + \frac{12 !}{1 ! ( 12 - 1 ) !} x^{11} (- \frac{1}{3 x})^{1} + \frac{12 !}{2 ! ( 12 - 2 ) !} x^{10} (- \frac{1}{3 x})^{2} + \frac{12 !}{3 ! ( 12 - 3 ) !} x^{9} (- \frac{1}{3 x})^{3} + \frac{12 !}{4 ! ( 12 - 4 ) !} x^{8} (- \frac{1}{3 x})^{4} + \frac{12 !}{5 ! ( 12 - 5 ) !} x^{7} (- \frac{1}{3 x})^{5} + \frac{12 !}{6 ! ( 12 - 6 ) !} x^{6} (- \frac{1}{3 x})^{6} + \frac{12 !}{7 ! ( 12 - 7 ) !} x^{5} (- \frac{1}{3 x})^{7} + \frac{12 !}{8 ! ( 12 - 8 ) !} x^{4} (- \frac{1}{3 x})^{8} + \frac{12 !}{9 ! ( 12 - 9 ) !} x^{3} (- \frac{1}{3 x})^{9} + \frac{12 !}{10 ! ( 12 - 10 ) !} x^{2} (- \frac{1}{3 x})^{10} + \frac{12 !}{11 ! ( 12 - 11 ) !} x^{1} (- \frac{1}{3 x})^{11} + \frac{12 !}{12 ! ( 12 - 12 ) !} x^{0} (- \frac{1}{3 x})^{12}

Vereinfache

\frac{12 !}{0 ! ( 12 - 0 ) !} x^{12} (- \frac{1}{3 x})^{0} : x^{12}

Vereinfache

\frac{12 !}{1 ! ( 12 - 1 ) !} x^{11} (- \frac{1}{3 x})^{1} : - 12 x^{\frac{32}{3}}

Vereinfache

\frac{12 !}{2 ! ( 12 - 2 ) !} x^{10} (- \frac{1}{3 x})^{2} : \frac{66 x ^{10}}{( 3 x ) ^{2}}

Vereinfache

\frac{12 !}{3 ! ( 12 - 3 ) !} x^{9} (- \frac{1}{3 x})^{3} : - 220 x^{8}

Vereinfache

\frac{12 !}{4 ! ( 12 - 4 ) !} x^{8} (- \frac{1}{3 x})^{4} : \frac{495 x ^{8}}{( 3 x ) ^{4}}

Vereinfache

\frac{12 !}{5 ! ( 12 - 5 ) !} x^{7} (- \frac{1}{3 x})^{5} : - \frac{792 x ^{7}}{( 3 x ) ^{5}}

Vereinfache

\frac{12 !}{6 ! ( 12 - 6 ) !} x^{6} (- \frac{1}{3 x})^{6} : 924 x^{4}

Vereinfache

\frac{12 !}{7 ! ( 12 - 7 ) !} x^{5} (- \frac{1}{3 x})^{7} : - \frac{792 x ^{5}}{( 3 x ) ^{7}}

Vereinfache

\frac{12 !}{8 ! ( 12 - 8 ) !} x^{4} (- \frac{1}{3 x})^{8} : \frac{495 x ^{4}}{( 3 x ) ^{8}}

Vereinfache

\frac{12 !}{9 ! ( 12 - 9 ) !} x^{3} (- \frac{1}{3 x})^{9} : - 220

Vereinfache

\frac{12 !}{10 ! ( 12 - 10 ) !} x^{2} (- \frac{1}{3 x})^{10} : \frac{66 x ^{2}}{( 3 x ) ^{10}}

Vereinfache

\frac{12 !}{11 ! ( 12 - 11 ) !} x^{1} (- \frac{1}{3 x})^{11} : - \frac{12 x}{( 3 x ) ^{11}}

Vereinfache

\frac{12 !}{12 ! ( 12 - 12 ) !} x^{0} (- \frac{1}{3 x})^{12} : \frac{1}{x ^{4}}

= x^{12} - 12 x^{\frac{32}{3}} + \frac{66 x ^{10}}{( 3 x ) ^{2}} - 220 x^{8} + \frac{495 x ^{8}}{( 3 x ) ^{4}} - \frac{792 x ^{7}}{( 3 x ) ^{5}} + 924 x^{4} - \frac{792 x ^{5}}{( 3 x ) ^{7}} + \frac{495 x ^{4}}{( 3 x ) ^{8}} - 220 + \frac{66 x ^{2}}{( 3 x ) ^{10}} - \frac{12 x}{( 3 x ) ^{11}} + \frac{1}{x ^{4}}

\frac{495 x ^{8}}{( 3 x ) ^{4}} = \frac{495 x ^{7}}{3 x}

\frac{792 x ^{7}}{( 3 x ) ^{5}} = \frac{792 x ^{6}}{( 3 x ) ^{2}}

\frac{792 x ^{5}}{( 3 x ) ^{7}} = \frac{792 x ^{3}}{3 x}

\frac{495 x ^{4}}{( 3 x ) ^{8}} = \frac{495 x ^{2}}{( 3 x ) ^{2}}

\frac{66 x ^{2}}{( 3 x ) ^{10}} = \frac{66}{x ^{\frac{4}{3}}}

\frac{12 x}{( 3 x ) ^{11}} = \frac{12}{x ^{\frac{8}{3}}}

= x^{12} - 12 x^{\frac{32}{3}} + \frac{66 x ^{10}}{( 3 x ) ^{2}} - 220 x^{8} + \frac{495 x ^{7}}{3 x} - \frac{792 x ^{6}}{( 3 x ) ^{2}} + 924 x^{4} - \frac{792 x ^{3}}{3 x} + \frac{495 x ^{2}}{( 3 x ) ^{2}} - 220 + \frac{66}{x ^{\frac{4}{3}}} - \frac{12}{x ^{\frac{8}{3}}} + \frac{1}{x ^{4}}

Füge zusammen:

x^{12} - 12 x^{\frac{32}{3}} + \frac{66 x ^{10}}{( 3 x ) ^{2}} - 220 x^{8} + \frac{495 x ^{7}}{3 x} - \frac{792 x ^{6}}{( 3 x ) ^{2}} + 924 x^{4} - \frac{792 x ^{3}}{3 x} + \frac{495 x ^{2}}{( 3 x ) ^{2}} - 220 + \frac{66}{x ^{\frac{4}{3}}} - \frac{12}{x ^{\frac{8}{3}}} + \frac{1}{x ^{4}} : x^{12} - 12 x^{\frac{32}{3}} + \frac{66 x ^{10} - 792 x ^{6} + 495 x ^{2}}{( 3 x ) ^{2}} - 220 x^{8} + 99 x^{2} 3 x^{2} (5 x^{4} - 8) + 924 x^{4} - 220 + \frac{66}{x ^{\frac{4}{3}}} - \frac{12}{x ^{\frac{8}{3}}} + \frac{1}{x ^{4}}

= x^{12} - 12 x^{\frac{32}{3}} + \frac{66 x ^{10} - 792 x ^{6} + 495 x ^{2}}{( 3 x ) ^{2}} - 220 x^{8} + 99 x^{2} 3 x^{2} (5 x^{4} - 8) + 924 x^{4} - 220 + \frac{66}{x ^{\frac{4}{3}}} - \frac{12}{x ^{\frac{8}{3}}} + \frac{1}{x ^{4}}

Schreibe

99 x^{2} 3 x^{2} (5 x^{4} - 8)

um:

495 x^{6} 3 x^{2} - 792 x^{2} 3 x^{2}

= x^{12} - 12 x^{\frac{32}{3}} + \frac{66 x ^{10} - 792 x ^{6} + 495 x ^{2}}{( 3 x ) ^{2}} - 220 x^{8} + 495 x^{6} 3 x^{2} - 792 x^{2} 3 x^{2} + 924 x^{4} - 220 + \frac{66}{x ^{\frac{4}{3}}} - \frac{12}{x ^{\frac{8}{3}}} + \frac{1}{x ^{4}}

Vereinfache

x^{12} - 12 x^{\frac{32}{3}} + \frac{66 x ^{10} - 792 x ^{6} + 495 x ^{2}}{( 3 x ) ^{2}} - 220 x^{8} + 495 x^{6} 3 x^{2} - 792 x^{2} 3 x^{2} + 924 x^{4} - 220 + \frac{66}{x ^{\frac{4}{3}}} - \frac{12}{x ^{\frac{8}{3}}} + \frac{1}{x ^{4}} : x^{12} - 12 3 x^{32} + \frac{66 x ^{10} - 792 x ^{6} + 495 x ^{2}}{( 3 x ) ^{2}} - 220 x^{8} + 495 x^{6} 3 x^{2} - 792 x^{2} 3 x^{2} + 924 x^{4} - 220 + \frac{66}{3 x ^{4}} - \frac{12}{3 x ^{8}} + \frac{1}{x ^{4}}

= x^{12} - 12 3 x^{32} + \frac{66 x ^{10} - 792 x ^{6} + 495 x ^{2}}{( 3 x ) ^{2}} - 220 x^{8} + 495 x^{6} 3 x^{2} - 792 x^{2} 3 x^{2} + 924 x^{4} - 220 + \frac{66}{3 x ^{4}} - \frac{12}{3 x ^{8}} + \frac{1}{x ^{4}}

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