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Popular Álgebra >

expandir (m+1)^{11}

Solução

expandir

(m + 1)^{11}

Solução

m^{11} + 11 m^{10} + 55 m^{9} + 165 m^{8} + 330 m^{7} + 462 m^{6} + 462 m^{5} + 330 m^{4} + 165 m^{3} + 55 m^{2} + 11 m + 1

Passos da solução

(m + 1)^{11}

Aplicar o teorema do binômio:

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

a = m, b = 1

= i = 0 \sum 11 (i 11) m^{(11 - i)} \cdot 1^{i}

Expandir a soma

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

Simplificar

\frac{11 !}{0 ! ( 11 - 0 ) !} m^{11} \cdot 1^{0} : m^{11}

Simplificar

\frac{11 !}{1 ! ( 11 - 1 ) !} m^{10} \cdot 1^{1} : 11 m^{10}

Simplificar

\frac{11 !}{2 ! ( 11 - 2 ) !} m^{9} \cdot 1^{2} : 55 m^{9}

Simplificar

\frac{11 !}{3 ! ( 11 - 3 ) !} m^{8} \cdot 1^{3} : 165 m^{8}

Simplificar

\frac{11 !}{4 ! ( 11 - 4 ) !} m^{7} \cdot 1^{4} : 330 m^{7}

Simplificar

\frac{11 !}{5 ! ( 11 - 5 ) !} m^{6} \cdot 1^{5} : 462 m^{6}

Simplificar

\frac{11 !}{6 ! ( 11 - 6 ) !} m^{5} \cdot 1^{6} : 462 m^{5}

Simplificar

\frac{11 !}{7 ! ( 11 - 7 ) !} m^{4} \cdot 1^{7} : 330 m^{4}

Simplificar

\frac{11 !}{8 ! ( 11 - 8 ) !} m^{3} \cdot 1^{8} : 165 m^{3}

Simplificar

\frac{11 !}{9 ! ( 11 - 9 ) !} m^{2} \cdot 1^{9} : 55 m^{2}

Simplificar

\frac{11 !}{10 ! ( 11 - 10 ) !} m^{1} \cdot 1^{10} : 11 m

Simplificar

\frac{11 !}{11 ! ( 11 - 11 ) !} m^{0} \cdot 1^{11} : 1

= m^{11} + 11 m^{10} + 55 m^{9} + 165 m^{8} + 330 m^{7} + 462 m^{6} + 462 m^{5} + 330 m^{4} + 165 m^{3} + 55 m^{2} + 11 m + 1

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