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Popular Álgebra >

simplificar (i+1)^{11}

Solução

simplificar

(i + 1)^{11}

Solução

- 32 + 32 i

Passos da solução

(i + 1)^{11}

Aplicar o teorema do binômio:

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

a = i, b = 1

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

Expandir a soma

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

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

Simplificar

i^{11} + 11 i^{10} + 55 i^{9} + 165 i^{8} + 330 i^{7} + 462 i^{6} + 462 i^{5} + 330 i^{4} + 165 i^{3} + 55 i^{2} + 11 i + 1 : 32 i - 32

= 32 i - 32

Rewrite in standard complex form:

- 32 + 32 i

= - 32 + 32 i

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