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Popular Álgebra >

expandir (x+4y)^{12}

Solução

expandir

(x + 4 y)^{12}

Solução

x^{12} + 48 x^{11} y + 1056 x^{10} y^{2} + 14080 x^{9} y^{3} + 126720 x^{8} y^{4} + 811008 x^{7} y^{5} + 3784704 x^{6} y^{6} + 12976128 x^{5} y^{7} + 32440320 x^{4} y^{8} + 57671680 x^{3} y^{9} + 69206016 x^{2} y^{10} + 50331648 x y^{11} + 16777216 y^{12}

Passos da solução

(x + 4 y)^{12}

Aplicar o teorema do binômio:

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

a = x, b = 4 y

= i = 0 \sum 12 (i 12) x^{(12 - i)} (4 y)^{i}

Expandir a soma

= \frac{12 !}{0 ! ( 12 - 0 ) !} x^{12} (4 y)^{0} + \frac{12 !}{1 ! ( 12 - 1 ) !} x^{11} (4 y)^{1} + \frac{12 !}{2 ! ( 12 - 2 ) !} x^{10} (4 y)^{2} + \frac{12 !}{3 ! ( 12 - 3 ) !} x^{9} (4 y)^{3} + \frac{12 !}{4 ! ( 12 - 4 ) !} x^{8} (4 y)^{4} + \frac{12 !}{5 ! ( 12 - 5 ) !} x^{7} (4 y)^{5} + \frac{12 !}{6 ! ( 12 - 6 ) !} x^{6} (4 y)^{6} + \frac{12 !}{7 ! ( 12 - 7 ) !} x^{5} (4 y)^{7} + \frac{12 !}{8 ! ( 12 - 8 ) !} x^{4} (4 y)^{8} + \frac{12 !}{9 ! ( 12 - 9 ) !} x^{3} (4 y)^{9} + \frac{12 !}{10 ! ( 12 - 10 ) !} x^{2} (4 y)^{10} + \frac{12 !}{11 ! ( 12 - 11 ) !} x^{1} (4 y)^{11} + \frac{12 !}{12 ! ( 12 - 12 ) !} x^{0} (4 y)^{12}

Simplificar

\frac{12 !}{0 ! ( 12 - 0 ) !} x^{12} (4 y)^{0} : x^{12}

Simplificar

\frac{12 !}{1 ! ( 12 - 1 ) !} x^{11} (4 y)^{1} : 48 x^{11} y

Simplificar

\frac{12 !}{2 ! ( 12 - 2 ) !} x^{10} (4 y)^{2} : 1056 x^{10} y^{2}

Simplificar

\frac{12 !}{3 ! ( 12 - 3 ) !} x^{9} (4 y)^{3} : 14080 x^{9} y^{3}

Simplificar

\frac{12 !}{4 ! ( 12 - 4 ) !} x^{8} (4 y)^{4} : 126720 x^{8} y^{4}

Simplificar

\frac{12 !}{5 ! ( 12 - 5 ) !} x^{7} (4 y)^{5} : 811008 x^{7} y^{5}

Simplificar

\frac{12 !}{6 ! ( 12 - 6 ) !} x^{6} (4 y)^{6} : 3784704 x^{6} y^{6}

Simplificar

\frac{12 !}{7 ! ( 12 - 7 ) !} x^{5} (4 y)^{7} : 12976128 x^{5} y^{7}

Simplificar

\frac{12 !}{8 ! ( 12 - 8 ) !} x^{4} (4 y)^{8} : 32440320 x^{4} y^{8}

Simplificar

\frac{12 !}{9 ! ( 12 - 9 ) !} x^{3} (4 y)^{9} : 57671680 x^{3} y^{9}

Simplificar

\frac{12 !}{10 ! ( 12 - 10 ) !} x^{2} (4 y)^{10} : 69206016 x^{2} y^{10}

Simplificar

\frac{12 !}{11 ! ( 12 - 11 ) !} x^{1} (4 y)^{11} : 50331648 x y^{11}

Simplificar

\frac{12 !}{12 ! ( 12 - 12 ) !} x^{0} (4 y)^{12} : 16777216 y^{12}

= x^{12} + 48 x^{11} y + 1056 x^{10} y^{2} + 14080 x^{9} y^{3} + 126720 x^{8} y^{4} + 811008 x^{7} y^{5} + 3784704 x^{6} y^{6} + 12976128 x^{5} y^{7} + 32440320 x^{4} y^{8} + 57671680 x^{3} y^{9} + 69206016 x^{2} y^{10} + 50331648 x y^{11} + 16777216 y^{12}

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