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Popolare Algebra >

espandere (1/(x^3)-x^2)^{18}

Soluzione

espandere

(\frac{1}{x ^{3}} - x^{2})^{18}

Soluzione

\frac{1}{x ^{54}} - \frac{18}{x ^{49}} + \frac{153}{x ^{44}} - \frac{816}{x ^{39}} + \frac{3060}{x ^{34}} - \frac{8568}{x ^{29}} + \frac{18564}{x ^{24}} - \frac{31824}{x ^{19}} + \frac{43758}{x ^{14}} - \frac{48620}{x ^{9}} + \frac{43758}{x ^{4}} - 31824 x + 18564 x^{6} - 8568 x^{11} + 3060 x^{16} - 816 x^{21} + 153 x^{26} - 18 x^{31} + x^{36}

Fasi della soluzione

(\frac{1}{x ^{3}} - x^{2})^{18}

Applicare il teorema binomiale:

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

a = \frac{1}{x ^{3}}, b = - x^{2}

= i = 0 \sum 18 (i 18) (\frac{1}{x ^{3}})^{(18 - i)} (- x^{2})^{i}

Espandere la sommatoria

= \frac{18 !}{0 ! ( 18 - 0 ) !} (\frac{1}{x ^{3}})^{18} (- x^{2})^{0} + \frac{18 !}{1 ! ( 18 - 1 ) !} (\frac{1}{x ^{3}})^{17} (- x^{2})^{1} + \frac{18 !}{2 ! ( 18 - 2 ) !} (\frac{1}{x ^{3}})^{16} (- x^{2})^{2} + \frac{18 !}{3 ! ( 18 - 3 ) !} (\frac{1}{x ^{3}})^{15} (- x^{2})^{3} + \frac{18 !}{4 ! ( 18 - 4 ) !} (\frac{1}{x ^{3}})^{14} (- x^{2})^{4} + \frac{18 !}{5 ! ( 18 - 5 ) !} (\frac{1}{x ^{3}})^{13} (- x^{2})^{5} + \frac{18 !}{6 ! ( 18 - 6 ) !} (\frac{1}{x ^{3}})^{12} (- x^{2})^{6} + \frac{18 !}{7 ! ( 18 - 7 ) !} (\frac{1}{x ^{3}})^{11} (- x^{2})^{7} + \frac{18 !}{8 ! ( 18 - 8 ) !} (\frac{1}{x ^{3}})^{10} (- x^{2})^{8} + \frac{18 !}{9 ! ( 18 - 9 ) !} (\frac{1}{x ^{3}})^{9} (- x^{2})^{9} + \frac{18 !}{10 ! ( 18 - 10 ) !} (\frac{1}{x ^{3}})^{8} (- x^{2})^{10} + \frac{18 !}{11 ! ( 18 - 11 ) !} (\frac{1}{x ^{3}})^{7} (- x^{2})^{11} + \frac{18 !}{12 ! ( 18 - 12 ) !} (\frac{1}{x ^{3}})^{6} (- x^{2})^{12} + \frac{18 !}{13 ! ( 18 - 13 ) !} (\frac{1}{x ^{3}})^{5} (- x^{2})^{13} + \frac{18 !}{14 ! ( 18 - 14 ) !} (\frac{1}{x ^{3}})^{4} (- x^{2})^{14} + \frac{18 !}{15 ! ( 18 - 15 ) !} (\frac{1}{x ^{3}})^{3} (- x^{2})^{15} + \frac{18 !}{16 ! ( 18 - 16 ) !} (\frac{1}{x ^{3}})^{2} (- x^{2})^{16} + \frac{18 !}{17 ! ( 18 - 17 ) !} (\frac{1}{x ^{3}})^{1} (- x^{2})^{17} + \frac{18 !}{18 ! ( 18 - 18 ) !} (\frac{1}{x ^{3}})^{0} (- x^{2})^{18}

Semplificare

\frac{18 !}{0 ! ( 18 - 0 ) !} (\frac{1}{x ^{3}})^{18} (- x^{2})^{0} : \frac{1}{x ^{54}}

Semplificare

\frac{18 !}{1 ! ( 18 - 1 ) !} (\frac{1}{x ^{3}})^{17} (- x^{2})^{1} : - \frac{18}{x ^{49}}

Semplificare

\frac{18 !}{2 ! ( 18 - 2 ) !} (\frac{1}{x ^{3}})^{16} (- x^{2})^{2} : \frac{153}{x ^{44}}

Semplificare

\frac{18 !}{3 ! ( 18 - 3 ) !} (\frac{1}{x ^{3}})^{15} (- x^{2})^{3} : - \frac{816}{x ^{39}}

Semplificare

\frac{18 !}{4 ! ( 18 - 4 ) !} (\frac{1}{x ^{3}})^{14} (- x^{2})^{4} : \frac{3060}{x ^{34}}

Semplificare

\frac{18 !}{5 ! ( 18 - 5 ) !} (\frac{1}{x ^{3}})^{13} (- x^{2})^{5} : - \frac{8568}{x ^{29}}

Semplificare

\frac{18 !}{6 ! ( 18 - 6 ) !} (\frac{1}{x ^{3}})^{12} (- x^{2})^{6} : \frac{18564}{x ^{24}}

Semplificare

\frac{18 !}{7 ! ( 18 - 7 ) !} (\frac{1}{x ^{3}})^{11} (- x^{2})^{7} : - \frac{31824}{x ^{19}}

Semplificare

\frac{18 !}{8 ! ( 18 - 8 ) !} (\frac{1}{x ^{3}})^{10} (- x^{2})^{8} : \frac{43758}{x ^{14}}

Semplificare

\frac{18 !}{9 ! ( 18 - 9 ) !} (\frac{1}{x ^{3}})^{9} (- x^{2})^{9} : - \frac{48620}{x ^{9}}

Semplificare

\frac{18 !}{10 ! ( 18 - 10 ) !} (\frac{1}{x ^{3}})^{8} (- x^{2})^{10} : \frac{43758}{x ^{4}}

Semplificare

\frac{18 !}{11 ! ( 18 - 11 ) !} (\frac{1}{x ^{3}})^{7} (- x^{2})^{11} : - 31824 x

Semplificare

\frac{18 !}{12 ! ( 18 - 12 ) !} (\frac{1}{x ^{3}})^{6} (- x^{2})^{12} : 18564 x^{6}

Semplificare

\frac{18 !}{13 ! ( 18 - 13 ) !} (\frac{1}{x ^{3}})^{5} (- x^{2})^{13} : - 8568 x^{11}

Semplificare

\frac{18 !}{14 ! ( 18 - 14 ) !} (\frac{1}{x ^{3}})^{4} (- x^{2})^{14} : 3060 x^{16}

Semplificare

\frac{18 !}{15 ! ( 18 - 15 ) !} (\frac{1}{x ^{3}})^{3} (- x^{2})^{15} : - 816 x^{21}

Semplificare

\frac{18 !}{16 ! ( 18 - 16 ) !} (\frac{1}{x ^{3}})^{2} (- x^{2})^{16} : 153 x^{26}

Semplificare

\frac{18 !}{17 ! ( 18 - 17 ) !} (\frac{1}{x ^{3}})^{1} (- x^{2})^{17} : - 18 x^{31}

Semplificare

\frac{18 !}{18 ! ( 18 - 18 ) !} (\frac{1}{x ^{3}})^{0} (- x^{2})^{18} : x^{36}

= \frac{1}{x ^{54}} - \frac{18}{x ^{49}} + \frac{153}{x ^{44}} - \frac{816}{x ^{39}} + \frac{3060}{x ^{34}} - \frac{8568}{x ^{29}} + \frac{18564}{x ^{24}} - \frac{31824}{x ^{19}} + \frac{43758}{x ^{14}} - \frac{48620}{x ^{9}} + \frac{43758}{x ^{4}} - 31824 x + 18564 x^{6} - 8568 x^{11} + 3060 x^{16} - 816 x^{21} + 153 x^{26} - 18 x^{31} + x^{36}

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