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Popular Álgebra >

expandir (a+b)^9

Solução

expandir

(a + b)^{9}

Solução

a^{9} + 9 a^{8} b + 36 a^{7} b^{2} + 84 a^{6} b^{3} + 126 a^{5} b^{4} + 126 a^{4} b^{5} + 84 a^{3} b^{6} + 36 a^{2} b^{7} + 9 a b^{8} + b^{9}

Passos da solução

(a + b)^{9}

Aplicar o teorema do binômio:

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

a = a, b = b

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

Expandir a soma

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

Simplificar

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

Simplificar

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

Simplificar

\frac{9 !}{2 ! ( 9 - 2 ) !} a^{7} b^{2} : 36 a^{7} b^{2}

Simplificar

\frac{9 !}{3 ! ( 9 - 3 ) !} a^{6} b^{3} : 84 a^{6} b^{3}

Simplificar

\frac{9 !}{4 ! ( 9 - 4 ) !} a^{5} b^{4} : 126 a^{5} b^{4}

Simplificar

\frac{9 !}{5 ! ( 9 - 5 ) !} a^{4} b^{5} : 126 a^{4} b^{5}

Simplificar

\frac{9 !}{6 ! ( 9 - 6 ) !} a^{3} b^{6} : 84 a^{3} b^{6}

Simplificar

\frac{9 !}{7 ! ( 9 - 7 ) !} a^{2} b^{7} : 36 a^{2} b^{7}

Simplificar

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

Simplificar

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

= a^{9} + 9 a^{8} b + 36 a^{7} b^{2} + 84 a^{6} b^{3} + 126 a^{5} b^{4} + 126 a^{4} b^{5} + 84 a^{3} b^{6} + 36 a^{2} b^{7} + 9 a b^{8} + b^{9}

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