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Popular Álgebra >

simplificar (1+2i)^{20}

Solución

simplificar

(1 + 2 i)^{20}

Solución

- 9653287 - 1476984 i

Pasos de solución

(1 + 2 i)^{20}

Aplicar el teorema del binomio:

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

a = 1, b = 2 i

= i = 0 \sum 20 (i 20) \cdot 1^{(20 - i)} (2 i)^{i}

Expandir sumatorio

= \frac{20 !}{0 ! ( 20 - 0 ) !} \cdot 1^{20} (2 i)^{0} + \frac{20 !}{1 ! ( 20 - 1 ) !} \cdot 1^{19} (2 i)^{1} + \frac{20 !}{2 ! ( 20 - 2 ) !} \cdot 1^{18} (2 i)^{2} + \frac{20 !}{3 ! ( 20 - 3 ) !} \cdot 1^{17} (2 i)^{3} + \frac{20 !}{4 ! ( 20 - 4 ) !} \cdot 1^{16} (2 i)^{4} + \frac{20 !}{5 ! ( 20 - 5 ) !} \cdot 1^{15} (2 i)^{5} + \frac{20 !}{6 ! ( 20 - 6 ) !} \cdot 1^{14} (2 i)^{6} + \frac{20 !}{7 ! ( 20 - 7 ) !} \cdot 1^{13} (2 i)^{7} + \frac{20 !}{8 ! ( 20 - 8 ) !} \cdot 1^{12} (2 i)^{8} + \frac{20 !}{9 ! ( 20 - 9 ) !} \cdot 1^{11} (2 i)^{9} + \frac{20 !}{10 ! ( 20 - 10 ) !} \cdot 1^{10} (2 i)^{10} + \frac{20 !}{11 ! ( 20 - 11 ) !} \cdot 1^{9} (2 i)^{11} + \frac{20 !}{12 ! ( 20 - 12 ) !} \cdot 1^{8} (2 i)^{12} + \frac{20 !}{13 ! ( 20 - 13 ) !} \cdot 1^{7} (2 i)^{13} + \frac{20 !}{14 ! ( 20 - 14 ) !} \cdot 1^{6} (2 i)^{14} + \frac{20 !}{15 ! ( 20 - 15 ) !} \cdot 1^{5} (2 i)^{15} + \frac{20 !}{16 ! ( 20 - 16 ) !} \cdot 1^{4} (2 i)^{16} + \frac{20 !}{17 ! ( 20 - 17 ) !} \cdot 1^{3} (2 i)^{17} + \frac{20 !}{18 ! ( 20 - 18 ) !} \cdot 1^{2} (2 i)^{18} + \frac{20 !}{19 ! ( 20 - 19 ) !} \cdot 1^{1} (2 i)^{19} + \frac{20 !}{20 ! ( 20 - 20 ) !} \cdot 1^{0} (2 i)^{20}

Simplificar

\frac{20 !}{0 ! ( 20 - 0 ) !} \cdot 1^{20} (2 i)^{0} : 1

Simplificar

\frac{20 !}{1 ! ( 20 - 1 ) !} \cdot 1^{19} (2 i)^{1} : 40 i

Simplificar

\frac{20 !}{2 ! ( 20 - 2 ) !} \cdot 1^{18} (2 i)^{2} : 760 i^{2}

Simplificar

\frac{20 !}{3 ! ( 20 - 3 ) !} \cdot 1^{17} (2 i)^{3} : 9120 i^{3}

Simplificar

\frac{20 !}{4 ! ( 20 - 4 ) !} \cdot 1^{16} (2 i)^{4} : 77520 i^{4}

Simplificar

\frac{20 !}{5 ! ( 20 - 5 ) !} \cdot 1^{15} (2 i)^{5} : 496128 i^{5}

Simplificar

\frac{20 !}{6 ! ( 20 - 6 ) !} \cdot 1^{14} (2 i)^{6} : 2480640 i^{6}

Simplificar

\frac{20 !}{7 ! ( 20 - 7 ) !} \cdot 1^{13} (2 i)^{7} : 9922560 i^{7}

Simplificar

\frac{20 !}{8 ! ( 20 - 8 ) !} \cdot 1^{12} (2 i)^{8} : 32248320 i^{8}

Simplificar

\frac{20 !}{9 ! ( 20 - 9 ) !} \cdot 1^{11} (2 i)^{9} : 85995520 i^{9}

Simplificar

\frac{20 !}{10 ! ( 20 - 10 ) !} \cdot 1^{10} (2 i)^{10} : 189190144 i^{10}

Simplificar

\frac{20 !}{11 ! ( 20 - 11 ) !} \cdot 1^{9} (2 i)^{11} : 343982080 i^{11}

Simplificar

\frac{20 !}{12 ! ( 20 - 12 ) !} \cdot 1^{8} (2 i)^{12} : 515973120 i^{12}

Simplificar

\frac{20 !}{13 ! ( 20 - 13 ) !} \cdot 1^{7} (2 i)^{13} : 635043840 i^{13}

Simplificar

\frac{20 !}{14 ! ( 20 - 14 ) !} \cdot 1^{6} (2 i)^{14} : 635043840 i^{14}

Simplificar

\frac{20 !}{15 ! ( 20 - 15 ) !} \cdot 1^{5} (2 i)^{15} : 508035072 i^{15}

Simplificar

\frac{20 !}{16 ! ( 20 - 16 ) !} \cdot 1^{4} (2 i)^{16} : 317521920 i^{16}

Simplificar

\frac{20 !}{17 ! ( 20 - 17 ) !} \cdot 1^{3} (2 i)^{17} : 149422080 i^{17}

Simplificar

\frac{20 !}{18 ! ( 20 - 18 ) !} \cdot 1^{2} (2 i)^{18} : 49807360 i^{18}

Simplificar

\frac{20 !}{19 ! ( 20 - 19 ) !} \cdot 1^{1} (2 i)^{19} : 10485760 i^{19}

Simplificar

\frac{20 !}{20 ! ( 20 - 20 ) !} \cdot 1^{0} (2 i)^{20} : 1048576 i^{20}

= 1 + 40 i + 760 i^{2} + 9120 i^{3} + 77520 i^{4} + 496128 i^{5} + 2480640 i^{6} + 9922560 i^{7} + 32248320 i^{8} + 85995520 i^{9} + 189190144 i^{10} + 343982080 i^{11} + 515973120 i^{12} + 635043840 i^{13} + 635043840 i^{14} + 508035072 i^{15} + 317521920 i^{16} + 149422080 i^{17} + 49807360 i^{18} + 10485760 i^{19} + 1048576 i^{20}

Simplificar

1 + 40 i + 760 i^{2} + 9120 i^{3} + 77520 i^{4} + 496128 i^{5} + 2480640 i^{6} + 9922560 i^{7} + 32248320 i^{8} + 85995520 i^{9} + 189190144 i^{10} + 343982080 i^{11} + 515973120 i^{12} + 635043840 i^{13} + 635043840 i^{14} + 508035072 i^{15} + 317521920 i^{16} + 149422080 i^{17} + 49807360 i^{18} + 10485760 i^{19} + 1048576 i^{20} : - 1476984 i - 9653287

= - 1476984 i - 9653287

Escribe en forma compleja estándar:

- 9653287 - 1476984 i

= - 9653287 - 1476984 i

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