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Popolare Algebra >

espandere (6x^2+5/x)^8

Soluzione

espandere

(6 x^{2} + \frac{5}{x})^{8}

Soluzione

1679616 x^{16} + 11197440 x^{13} + 32659200 x^{10} + 54432000 x^{7} + 56700000 x^{4} + 37800000 x + \frac{15750000}{x ^{2}} + \frac{3750000}{x ^{5}} + \frac{390625}{x ^{8}}

Fasi della soluzione

(6 x^{2} + \frac{5}{x})^{8}

Applicare il teorema binomiale:

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

a = 6 x^{2}, b = \frac{5}{x}

= i = 0 \sum 8 (i 8) (6 x^{2})^{(8 - i)} (\frac{5}{x})^{i}

Espandere la sommatoria

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

Semplificare

\frac{8 !}{0 ! ( 8 - 0 ) !} (6 x^{2})^{8} (\frac{5}{x})^{0} : 1679616 x^{16}

Semplificare

\frac{8 !}{1 ! ( 8 - 1 ) !} (6 x^{2})^{7} (\frac{5}{x})^{1} : 11197440 x^{13}

Semplificare

\frac{8 !}{2 ! ( 8 - 2 ) !} (6 x^{2})^{6} (\frac{5}{x})^{2} : 32659200 x^{10}

Semplificare

\frac{8 !}{3 ! ( 8 - 3 ) !} (6 x^{2})^{5} (\frac{5}{x})^{3} : 54432000 x^{7}

Semplificare

\frac{8 !}{4 ! ( 8 - 4 ) !} (6 x^{2})^{4} (\frac{5}{x})^{4} : 56700000 x^{4}

Semplificare

\frac{8 !}{5 ! ( 8 - 5 ) !} (6 x^{2})^{3} (\frac{5}{x})^{5} : 37800000 x

Semplificare

\frac{8 !}{6 ! ( 8 - 6 ) !} (6 x^{2})^{2} (\frac{5}{x})^{6} : \frac{15750000}{x ^{2}}

Semplificare

\frac{8 !}{7 ! ( 8 - 7 ) !} (6 x^{2})^{1} (\frac{5}{x})^{7} : \frac{3750000}{x ^{5}}

Semplificare

\frac{8 !}{8 ! ( 8 - 8 ) !} (6 x^{2})^{0} (\frac{5}{x})^{8} : \frac{390625}{x ^{8}}

= 1679616 x^{16} + 11197440 x^{13} + 32659200 x^{10} + 54432000 x^{7} + 56700000 x^{4} + 37800000 x + \frac{15750000}{x ^{2}} + \frac{3750000}{x ^{5}} + \frac{390625}{x ^{8}}

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