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Popolare Algebra >

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

Soluzione

espandere

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

Soluzione

512 x^{18} + 2304 x^{15} + 4608 x^{12} + 5376 x^{9} + 4032 x^{6} + 2016 x^{3} + 672 + \frac{144}{x ^{3}} + \frac{18}{x ^{6}} + \frac{1}{x ^{9}}

Fasi della soluzione

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

Applicare il teorema binomiale:

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

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

= i = 0 \sum 9 (i 9) (2 x^{2})^{(9 - i)} (\frac{1}{x})^{i}

Espandere la sommatoria

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

Semplificare

\frac{9 !}{0 ! ( 9 - 0 ) !} (2 x^{2})^{9} (\frac{1}{x})^{0} : 512 x^{18}

Semplificare

\frac{9 !}{1 ! ( 9 - 1 ) !} (2 x^{2})^{8} (\frac{1}{x})^{1} : 2304 x^{15}

Semplificare

\frac{9 !}{2 ! ( 9 - 2 ) !} (2 x^{2})^{7} (\frac{1}{x})^{2} : 4608 x^{12}

Semplificare

\frac{9 !}{3 ! ( 9 - 3 ) !} (2 x^{2})^{6} (\frac{1}{x})^{3} : 5376 x^{9}

Semplificare

\frac{9 !}{4 ! ( 9 - 4 ) !} (2 x^{2})^{5} (\frac{1}{x})^{4} : 4032 x^{6}

Semplificare

\frac{9 !}{5 ! ( 9 - 5 ) !} (2 x^{2})^{4} (\frac{1}{x})^{5} : 2016 x^{3}

Semplificare

\frac{9 !}{6 ! ( 9 - 6 ) !} (2 x^{2})^{3} (\frac{1}{x})^{6} : 672

Semplificare

\frac{9 !}{7 ! ( 9 - 7 ) !} (2 x^{2})^{2} (\frac{1}{x})^{7} : \frac{144}{x ^{3}}

Semplificare

\frac{9 !}{8 ! ( 9 - 8 ) !} (2 x^{2})^{1} (\frac{1}{x})^{8} : \frac{18}{x ^{6}}

Semplificare

\frac{9 !}{9 ! ( 9 - 9 ) !} (2 x^{2})^{0} (\frac{1}{x})^{9} : \frac{1}{x ^{9}}

= 512 x^{18} + 2304 x^{15} + 4608 x^{12} + 5376 x^{9} + 4032 x^{6} + 2016 x^{3} + 672 + \frac{144}{x ^{3}} + \frac{18}{x ^{6}} + \frac{1}{x ^{9}}

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