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Popolare Algebra >

espandere (a+2b)^{10}

Soluzione

espandere

(a + 2 b)^{10}

Soluzione

a^{10} + 20 a^{9} b + 180 a^{8} b^{2} + 960 a^{7} b^{3} + 3360 a^{6} b^{4} + 8064 a^{5} b^{5} + 13440 a^{4} b^{6} + 15360 a^{3} b^{7} + 11520 a^{2} b^{8} + 5120 a b^{9} + 1024 b^{10}

Fasi della soluzione

(a + 2 b)^{10}

Applicare il teorema binomiale:

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

a = a, b = 2 b

= i = 0 \sum 10 (i 10) a^{(10 - i)} (2 b)^{i}

Espandere la sommatoria

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

Semplificare

\frac{10 !}{0 ! ( 10 - 0 ) !} a^{10} (2 b)^{0} : a^{10}

Semplificare

\frac{10 !}{1 ! ( 10 - 1 ) !} a^{9} (2 b)^{1} : 20 a^{9} b

Semplificare

\frac{10 !}{2 ! ( 10 - 2 ) !} a^{8} (2 b)^{2} : 180 a^{8} b^{2}

Semplificare

\frac{10 !}{3 ! ( 10 - 3 ) !} a^{7} (2 b)^{3} : 960 a^{7} b^{3}

Semplificare

\frac{10 !}{4 ! ( 10 - 4 ) !} a^{6} (2 b)^{4} : 3360 a^{6} b^{4}

Semplificare

\frac{10 !}{5 ! ( 10 - 5 ) !} a^{5} (2 b)^{5} : 8064 a^{5} b^{5}

Semplificare

\frac{10 !}{6 ! ( 10 - 6 ) !} a^{4} (2 b)^{6} : 13440 a^{4} b^{6}

Semplificare

\frac{10 !}{7 ! ( 10 - 7 ) !} a^{3} (2 b)^{7} : 15360 a^{3} b^{7}

Semplificare

\frac{10 !}{8 ! ( 10 - 8 ) !} a^{2} (2 b)^{8} : 11520 a^{2} b^{8}

Semplificare

\frac{10 !}{9 ! ( 10 - 9 ) !} a^{1} (2 b)^{9} : 5120 a b^{9}

Semplificare

\frac{10 !}{10 ! ( 10 - 10 ) !} a^{0} (2 b)^{10} : 1024 b^{10}

= a^{10} + 20 a^{9} b + 180 a^{8} b^{2} + 960 a^{7} b^{3} + 3360 a^{6} b^{4} + 8064 a^{5} b^{5} + 13440 a^{4} b^{6} + 15360 a^{3} b^{7} + 11520 a^{2} b^{8} + 5120 a b^{9} + 1024 b^{10}

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