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Popular Álgebra >

expandir (e^x+1)^{12}

Solução

expandir

(e^{x} + 1)^{12}

Solução

e^{12 x} + 12 e^{11 x} + 66 e^{10 x} + 220 e^{9 x} + 495 e^{8 x} + 792 e^{7 x} + 924 e^{6 x} + 792 e^{5 x} + 495 e^{4 x} + 220 e^{3 x} + 66 e^{2 x} + 12 e^{x} + 1

Passos da solução

(e^{x} + 1)^{12}

Aplicar o teorema do binômio:

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

a = e^{x}, b = 1

= i = 0 \sum 12 (i 12) (e^{x})^{(12 - i)} \cdot 1^{i}

Expandir a soma

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

Simplificar

\frac{12 !}{0 ! ( 12 - 0 ) !} (e^{x})^{12} \cdot 1^{0} : e^{12 x}

Simplificar

\frac{12 !}{1 ! ( 12 - 1 ) !} (e^{x})^{11} \cdot 1^{1} : 12 e^{11 x}

Simplificar

\frac{12 !}{2 ! ( 12 - 2 ) !} (e^{x})^{10} \cdot 1^{2} : 66 e^{10 x}

Simplificar

\frac{12 !}{3 ! ( 12 - 3 ) !} (e^{x})^{9} \cdot 1^{3} : 220 e^{9 x}

Simplificar

\frac{12 !}{4 ! ( 12 - 4 ) !} (e^{x})^{8} \cdot 1^{4} : 495 e^{8 x}

Simplificar

\frac{12 !}{5 ! ( 12 - 5 ) !} (e^{x})^{7} \cdot 1^{5} : 792 e^{7 x}

Simplificar

\frac{12 !}{6 ! ( 12 - 6 ) !} (e^{x})^{6} \cdot 1^{6} : 924 e^{6 x}

Simplificar

\frac{12 !}{7 ! ( 12 - 7 ) !} (e^{x})^{5} \cdot 1^{7} : 792 e^{5 x}

Simplificar

\frac{12 !}{8 ! ( 12 - 8 ) !} (e^{x})^{4} \cdot 1^{8} : 495 e^{4 x}

Simplificar

\frac{12 !}{9 ! ( 12 - 9 ) !} (e^{x})^{3} \cdot 1^{9} : 220 e^{3 x}

Simplificar

\frac{12 !}{10 ! ( 12 - 10 ) !} (e^{x})^{2} \cdot 1^{10} : 66 e^{2 x}

Simplificar

\frac{12 !}{11 ! ( 12 - 11 ) !} (e^{x})^{1} \cdot 1^{11} : 12 e^{x}

\frac{12 !}{12 ! ( 12 - 12 ) !} (e^{x})^{0} \cdot 1^{12} = 1

= e^{12 x} + 12 e^{11 x} + 66 e^{10 x} + 220 e^{9 x} + 495 e^{8 x} + 792 e^{7 x} + 924 e^{6 x} + 792 e^{5 x} + 495 e^{4 x} + 220 e^{3 x} + 66 e^{2 x} + 12 e^{x} + 1

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