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Popolare Algebra >

espandere (a^2+2b)^9

Soluzione

espandere

(a^{2} + 2 b)^{9}

Soluzione

a^{18} + 18 a^{16} b + 144 a^{14} b^{2} + 672 a^{12} b^{3} + 2016 a^{10} b^{4} + 4032 a^{8} b^{5} + 5376 a^{6} b^{6} + 4608 a^{4} b^{7} + 2304 a^{2} b^{8} + 512 b^{9}

Fasi della soluzione

(a^{2} + 2 b)^{9}

Applicare il teorema binomiale:

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

a = a^{2}, b = 2 b

= i = 0 \sum 9 (i 9) (a^{2})^{(9 - i)} (2 b)^{i}

Espandere la sommatoria

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

Semplificare

\frac{9 !}{0 ! ( 9 - 0 ) !} (a^{2})^{9} (2 b)^{0} : a^{18}

Semplificare

\frac{9 !}{1 ! ( 9 - 1 ) !} (a^{2})^{8} (2 b)^{1} : 18 a^{16} b

Semplificare

\frac{9 !}{2 ! ( 9 - 2 ) !} (a^{2})^{7} (2 b)^{2} : 144 a^{14} b^{2}

Semplificare

\frac{9 !}{3 ! ( 9 - 3 ) !} (a^{2})^{6} (2 b)^{3} : 672 a^{12} b^{3}

Semplificare

\frac{9 !}{4 ! ( 9 - 4 ) !} (a^{2})^{5} (2 b)^{4} : 2016 a^{10} b^{4}

Semplificare

\frac{9 !}{5 ! ( 9 - 5 ) !} (a^{2})^{4} (2 b)^{5} : 4032 a^{8} b^{5}

Semplificare

\frac{9 !}{6 ! ( 9 - 6 ) !} (a^{2})^{3} (2 b)^{6} : 5376 a^{6} b^{6}

Semplificare

\frac{9 !}{7 ! ( 9 - 7 ) !} (a^{2})^{2} (2 b)^{7} : 4608 a^{4} b^{7}

Semplificare

\frac{9 !}{8 ! ( 9 - 8 ) !} (a^{2})^{1} (2 b)^{8} : 2304 a^{2} b^{8}

Semplificare

\frac{9 !}{9 ! ( 9 - 9 ) !} (a^{2})^{0} (2 b)^{9} : 512 b^{9}

= a^{18} + 18 a^{16} b + 144 a^{14} b^{2} + 672 a^{12} b^{3} + 2016 a^{10} b^{4} + 4032 a^{8} b^{5} + 5376 a^{6} b^{6} + 4608 a^{4} b^{7} + 2304 a^{2} b^{8} + 512 b^{9}

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