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Popular Álgebra >

simplificar (1+i)^{13}

Solução

simplificar

(1 + i)^{13}

Solução

- 64 - 64 i

Passos da solução

(1 + i)^{13}

Aplicar o teorema do binômio:

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

a = 1, b = i

= i = 0 \sum 13 (i 13) \cdot 1^{(13 - i)} i^{i}

Expandir a soma

= \frac{13 !}{0 ! ( 13 - 0 ) !} \cdot 1^{13} i^{0} + \frac{13 !}{1 ! ( 13 - 1 ) !} \cdot 1^{12} i^{1} + \frac{13 !}{2 ! ( 13 - 2 ) !} \cdot 1^{11} i^{2} + \frac{13 !}{3 ! ( 13 - 3 ) !} \cdot 1^{10} i^{3} + \frac{13 !}{4 ! ( 13 - 4 ) !} \cdot 1^{9} i^{4} + \frac{13 !}{5 ! ( 13 - 5 ) !} \cdot 1^{8} i^{5} + \frac{13 !}{6 ! ( 13 - 6 ) !} \cdot 1^{7} i^{6} + \frac{13 !}{7 ! ( 13 - 7 ) !} \cdot 1^{6} i^{7} + \frac{13 !}{8 ! ( 13 - 8 ) !} \cdot 1^{5} i^{8} + \frac{13 !}{9 ! ( 13 - 9 ) !} \cdot 1^{4} i^{9} + \frac{13 !}{10 ! ( 13 - 10 ) !} \cdot 1^{3} i^{10} + \frac{13 !}{11 ! ( 13 - 11 ) !} \cdot 1^{2} i^{11} + \frac{13 !}{12 ! ( 13 - 12 ) !} \cdot 1^{1} i^{12} + \frac{13 !}{13 ! ( 13 - 13 ) !} \cdot 1^{0} i^{13}

Simplificar

\frac{13 !}{0 ! ( 13 - 0 ) !} \cdot 1^{13} i^{0} : 1

Simplificar

\frac{13 !}{1 ! ( 13 - 1 ) !} \cdot 1^{12} i^{1} : 13 i

Simplificar

\frac{13 !}{2 ! ( 13 - 2 ) !} \cdot 1^{11} i^{2} : 78 i^{2}

Simplificar

\frac{13 !}{3 ! ( 13 - 3 ) !} \cdot 1^{10} i^{3} : 286 i^{3}

Simplificar

\frac{13 !}{4 ! ( 13 - 4 ) !} \cdot 1^{9} i^{4} : 715 i^{4}

Simplificar

\frac{13 !}{5 ! ( 13 - 5 ) !} \cdot 1^{8} i^{5} : 1287 i^{5}

Simplificar

\frac{13 !}{6 ! ( 13 - 6 ) !} \cdot 1^{7} i^{6} : 1716 i^{6}

Simplificar

\frac{13 !}{7 ! ( 13 - 7 ) !} \cdot 1^{6} i^{7} : 1716 i^{7}

Simplificar

\frac{13 !}{8 ! ( 13 - 8 ) !} \cdot 1^{5} i^{8} : 1287 i^{8}

Simplificar

\frac{13 !}{9 ! ( 13 - 9 ) !} \cdot 1^{4} i^{9} : 715 i^{9}

Simplificar

\frac{13 !}{10 ! ( 13 - 10 ) !} \cdot 1^{3} i^{10} : 286 i^{10}

Simplificar

\frac{13 !}{11 ! ( 13 - 11 ) !} \cdot 1^{2} i^{11} : 78 i^{11}

Simplificar

\frac{13 !}{12 ! ( 13 - 12 ) !} \cdot 1^{1} i^{12} : 13 i^{12}

Simplificar

\frac{13 !}{13 ! ( 13 - 13 ) !} \cdot 1^{0} i^{13} : i^{13}

= 1 + 13 i + 78 i^{2} + 286 i^{3} + 715 i^{4} + 1287 i^{5} + 1716 i^{6} + 1716 i^{7} + 1287 i^{8} + 715 i^{9} + 286 i^{10} + 78 i^{11} + 13 i^{12} + i^{13}

Simplificar

1 + 13 i + 78 i^{2} + 286 i^{3} + 715 i^{4} + 1287 i^{5} + 1716 i^{6} + 1716 i^{7} + 1287 i^{8} + 715 i^{9} + 286 i^{10} + 78 i^{11} + 13 i^{12} + i^{13} : - 64 i - 64

= - 64 i - 64

Rewrite in standard complex form:

- 64 - 64 i

= - 64 - 64 i

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