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Beliebt Algebra >

erweitern (x^2+3/(x^3))^{14}

Lösung

erweitern

(x^{2} + \frac{3}{x ^{3}})^{14}

Lösung

x^{28} + 42 x^{23} + 819 x^{18} + 9828 x^{13} + 81081 x^{8} + 486486 x^{3} + \frac{2189187}{x ^{2}} + \frac{7505784}{x ^{7}} + \frac{19702683}{x ^{12}} + \frac{39405366}{x ^{17}} + \frac{59108049}{x ^{22}} + \frac{64481508}{x ^{27}} + \frac{48361131}{x ^{32}} + \frac{22320522}{x ^{37}} + \frac{4782969}{x ^{42}}

Schritte zur Lösung

(x^{2} + \frac{3}{x ^{3}})^{14}

Wende Binomialsetz an:

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

a = x^{2}, b = \frac{3}{x ^{3}}

= i = 0 \sum 14 (i 14) (x^{2})^{(14 - i)} (\frac{3}{x ^{3}})^{i}

Erweitere Summation

= \frac{14 !}{0 ! ( 14 - 0 ) !} (x^{2})^{14} (\frac{3}{x ^{3}})^{0} + \frac{14 !}{1 ! ( 14 - 1 ) !} (x^{2})^{13} (\frac{3}{x ^{3}})^{1} + \frac{14 !}{2 ! ( 14 - 2 ) !} (x^{2})^{12} (\frac{3}{x ^{3}})^{2} + \frac{14 !}{3 ! ( 14 - 3 ) !} (x^{2})^{11} (\frac{3}{x ^{3}})^{3} + \frac{14 !}{4 ! ( 14 - 4 ) !} (x^{2})^{10} (\frac{3}{x ^{3}})^{4} + \frac{14 !}{5 ! ( 14 - 5 ) !} (x^{2})^{9} (\frac{3}{x ^{3}})^{5} + \frac{14 !}{6 ! ( 14 - 6 ) !} (x^{2})^{8} (\frac{3}{x ^{3}})^{6} + \frac{14 !}{7 ! ( 14 - 7 ) !} (x^{2})^{7} (\frac{3}{x ^{3}})^{7} + \frac{14 !}{8 ! ( 14 - 8 ) !} (x^{2})^{6} (\frac{3}{x ^{3}})^{8} + \frac{14 !}{9 ! ( 14 - 9 ) !} (x^{2})^{5} (\frac{3}{x ^{3}})^{9} + \frac{14 !}{10 ! ( 14 - 10 ) !} (x^{2})^{4} (\frac{3}{x ^{3}})^{10} + \frac{14 !}{11 ! ( 14 - 11 ) !} (x^{2})^{3} (\frac{3}{x ^{3}})^{11} + \frac{14 !}{12 ! ( 14 - 12 ) !} (x^{2})^{2} (\frac{3}{x ^{3}})^{12} + \frac{14 !}{13 ! ( 14 - 13 ) !} (x^{2})^{1} (\frac{3}{x ^{3}})^{13} + \frac{14 !}{14 ! ( 14 - 14 ) !} (x^{2})^{0} (\frac{3}{x ^{3}})^{14}

Vereinfache

\frac{14 !}{0 ! ( 14 - 0 ) !} (x^{2})^{14} (\frac{3}{x ^{3}})^{0} : x^{28}

Vereinfache

\frac{14 !}{1 ! ( 14 - 1 ) !} (x^{2})^{13} (\frac{3}{x ^{3}})^{1} : 42 x^{23}

Vereinfache

\frac{14 !}{2 ! ( 14 - 2 ) !} (x^{2})^{12} (\frac{3}{x ^{3}})^{2} : 819 x^{18}

Vereinfache

\frac{14 !}{3 ! ( 14 - 3 ) !} (x^{2})^{11} (\frac{3}{x ^{3}})^{3} : 9828 x^{13}

Vereinfache

\frac{14 !}{4 ! ( 14 - 4 ) !} (x^{2})^{10} (\frac{3}{x ^{3}})^{4} : 81081 x^{8}

Vereinfache

\frac{14 !}{5 ! ( 14 - 5 ) !} (x^{2})^{9} (\frac{3}{x ^{3}})^{5} : 486486 x^{3}

Vereinfache

\frac{14 !}{6 ! ( 14 - 6 ) !} (x^{2})^{8} (\frac{3}{x ^{3}})^{6} : \frac{2189187}{x ^{2}}

Vereinfache

\frac{14 !}{7 ! ( 14 - 7 ) !} (x^{2})^{7} (\frac{3}{x ^{3}})^{7} : \frac{7505784}{x ^{7}}

Vereinfache

\frac{14 !}{8 ! ( 14 - 8 ) !} (x^{2})^{6} (\frac{3}{x ^{3}})^{8} : \frac{19702683}{x ^{12}}

Vereinfache

\frac{14 !}{9 ! ( 14 - 9 ) !} (x^{2})^{5} (\frac{3}{x ^{3}})^{9} : \frac{39405366}{x ^{17}}

Vereinfache

\frac{14 !}{10 ! ( 14 - 10 ) !} (x^{2})^{4} (\frac{3}{x ^{3}})^{10} : \frac{59108049}{x ^{22}}

Vereinfache

\frac{14 !}{11 ! ( 14 - 11 ) !} (x^{2})^{3} (\frac{3}{x ^{3}})^{11} : \frac{64481508}{x ^{27}}

Vereinfache

\frac{14 !}{12 ! ( 14 - 12 ) !} (x^{2})^{2} (\frac{3}{x ^{3}})^{12} : \frac{48361131}{x ^{32}}

Vereinfache

\frac{14 !}{13 ! ( 14 - 13 ) !} (x^{2})^{1} (\frac{3}{x ^{3}})^{13} : \frac{22320522}{x ^{37}}

Vereinfache

\frac{14 !}{14 ! ( 14 - 14 ) !} (x^{2})^{0} (\frac{3}{x ^{3}})^{14} : \frac{4782969}{x ^{42}}

= x^{28} + 42 x^{23} + 819 x^{18} + 9828 x^{13} + 81081 x^{8} + 486486 x^{3} + \frac{2189187}{x ^{2}} + \frac{7505784}{x ^{7}} + \frac{19702683}{x ^{12}} + \frac{39405366}{x ^{17}} + \frac{59108049}{x ^{22}} + \frac{64481508}{x ^{27}} + \frac{48361131}{x ^{32}} + \frac{22320522}{x ^{37}} + \frac{4782969}{x ^{42}}

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