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Popolare Algebra >

espandere (2x^2+3/x)^{10}

Soluzione

espandere

(2 x^{2} + \frac{3}{x})^{10}

Soluzione

1024 x^{20} + 15360 x^{17} + 103680 x^{14} + 414720 x^{11} + 1088640 x^{8} + 1959552 x^{5} + 2449440 x^{2} + \frac{2099520}{x} + \frac{1180980}{x ^{4}} + \frac{393660}{x ^{7}} + \frac{59049}{x ^{10}}

Fasi della soluzione

(2 x^{2} + \frac{3}{x})^{10}

Applicare il teorema binomiale:

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

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

= i = 0 \sum 10 (i 10) (2 x^{2})^{(10 - i)} (\frac{3}{x})^{i}

Espandere la sommatoria

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

Semplificare

\frac{10 !}{0 ! ( 10 - 0 ) !} (2 x^{2})^{10} (\frac{3}{x})^{0} : 1024 x^{20}

Semplificare

\frac{10 !}{1 ! ( 10 - 1 ) !} (2 x^{2})^{9} (\frac{3}{x})^{1} : 15360 x^{17}

Semplificare

\frac{10 !}{2 ! ( 10 - 2 ) !} (2 x^{2})^{8} (\frac{3}{x})^{2} : 103680 x^{14}

Semplificare

\frac{10 !}{3 ! ( 10 - 3 ) !} (2 x^{2})^{7} (\frac{3}{x})^{3} : 414720 x^{11}

Semplificare

\frac{10 !}{4 ! ( 10 - 4 ) !} (2 x^{2})^{6} (\frac{3}{x})^{4} : 1088640 x^{8}

Semplificare

\frac{10 !}{5 ! ( 10 - 5 ) !} (2 x^{2})^{5} (\frac{3}{x})^{5} : 1959552 x^{5}

Semplificare

\frac{10 !}{6 ! ( 10 - 6 ) !} (2 x^{2})^{4} (\frac{3}{x})^{6} : 2449440 x^{2}

Semplificare

\frac{10 !}{7 ! ( 10 - 7 ) !} (2 x^{2})^{3} (\frac{3}{x})^{7} : \frac{2099520}{x}

Semplificare

\frac{10 !}{8 ! ( 10 - 8 ) !} (2 x^{2})^{2} (\frac{3}{x})^{8} : \frac{1180980}{x ^{4}}

Semplificare

\frac{10 !}{9 ! ( 10 - 9 ) !} (2 x^{2})^{1} (\frac{3}{x})^{9} : \frac{393660}{x ^{7}}

Semplificare

\frac{10 !}{10 ! ( 10 - 10 ) !} (2 x^{2})^{0} (\frac{3}{x})^{10} : \frac{59049}{x ^{10}}

= 1024 x^{20} + 15360 x^{17} + 103680 x^{14} + 414720 x^{11} + 1088640 x^{8} + 1959552 x^{5} + 2449440 x^{2} + \frac{2099520}{x} + \frac{1180980}{x ^{4}} + \frac{393660}{x ^{7}} + \frac{59049}{x ^{10}}

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