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Popolare Algebra >

espandere (1/(3x)+(-3)x^2)^8

Soluzione

espandere

(\frac{1}{3 x} + (- 3) x^{2})^{8}

Soluzione

\frac{1}{6561 x ^{8}} - \frac{8}{729 x ^{5}} + \frac{28}{81 x ^{2}} - \frac{56 x}{9} + 70 x^{4} - 504 x^{7} + 2268 x^{10} - 5832 x^{13} + 6561 x^{16}

Fasi della soluzione

(\frac{1}{3 x} + (- 3) x^{2})^{8}

Semplificare

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

= (\frac{1}{3 x} - 3 x^{2})^{8}

Applicare il teorema binomiale:

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

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

= i = 0 \sum 8 (i 8) (\frac{1}{3 x})^{(8 - i)} (- 3 x^{2})^{i}

Espandere la sommatoria

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

Semplificare

\frac{8 !}{0 ! ( 8 - 0 ) !} (\frac{1}{3 x})^{8} (- 3 x^{2})^{0} : \frac{1}{6561 x ^{8}}

Semplificare

\frac{8 !}{1 ! ( 8 - 1 ) !} (\frac{1}{3 x})^{7} (- 3 x^{2})^{1} : - \frac{8}{729 x ^{5}}

Semplificare

\frac{8 !}{2 ! ( 8 - 2 ) !} (\frac{1}{3 x})^{6} (- 3 x^{2})^{2} : \frac{28}{81 x ^{2}}

Semplificare

\frac{8 !}{3 ! ( 8 - 3 ) !} (\frac{1}{3 x})^{5} (- 3 x^{2})^{3} : - \frac{56 x}{9}

Semplificare

\frac{8 !}{4 ! ( 8 - 4 ) !} (\frac{1}{3 x})^{4} (- 3 x^{2})^{4} : 70 x^{4}

Semplificare

\frac{8 !}{5 ! ( 8 - 5 ) !} (\frac{1}{3 x})^{3} (- 3 x^{2})^{5} : - 504 x^{7}

Semplificare

\frac{8 !}{6 ! ( 8 - 6 ) !} (\frac{1}{3 x})^{2} (- 3 x^{2})^{6} : 2268 x^{10}

Semplificare

\frac{8 !}{7 ! ( 8 - 7 ) !} (\frac{1}{3 x})^{1} (- 3 x^{2})^{7} : - 5832 x^{13}

Semplificare

\frac{8 !}{8 ! ( 8 - 8 ) !} (\frac{1}{3 x})^{0} (- 3 x^{2})^{8} : 6561 x^{16}

= \frac{1}{6561 x ^{8}} - \frac{8}{729 x ^{5}} + \frac{28}{81 x ^{2}} - \frac{56 x}{9} + 70 x^{4} - 504 x^{7} + 2268 x^{10} - 5832 x^{13} + 6561 x^{16}

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