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Popular Álgebra >

simplificar (sqrt(3)+i)^{26}

Solución

simplificar

(3 + i)^{26}

Solución

33554432 + 335544323 i

Pasos de solución

(3 + i)^{26}

Aplicar el teorema del binomio:

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

a = 3, b = i

= i = 0 \sum 26 (i 26) (3)^{(26 - i)} i^{i}

Expandir sumatorio

= \frac{26 !}{0 ! ( 26 - 0 ) !} (3)^{26} i^{0} + \frac{26 !}{1 ! ( 26 - 1 ) !} (3)^{25} i^{1} + \frac{26 !}{2 ! ( 26 - 2 ) !} (3)^{24} i^{2} + \frac{26 !}{3 ! ( 26 - 3 ) !} (3)^{23} i^{3} + \frac{26 !}{4 ! ( 26 - 4 ) !} (3)^{22} i^{4} + \frac{26 !}{5 ! ( 26 - 5 ) !} (3)^{21} i^{5} + \frac{26 !}{6 ! ( 26 - 6 ) !} (3)^{20} i^{6} + \frac{26 !}{7 ! ( 26 - 7 ) !} (3)^{19} i^{7} + \frac{26 !}{8 ! ( 26 - 8 ) !} (3)^{18} i^{8} + \frac{26 !}{9 ! ( 26 - 9 ) !} (3)^{17} i^{9} + \frac{26 !}{10 ! ( 26 - 10 ) !} (3)^{16} i^{10} + \frac{26 !}{11 ! ( 26 - 11 ) !} (3)^{15} i^{11} + \frac{26 !}{12 ! ( 26 - 12 ) !} (3)^{14} i^{12} + \frac{26 !}{13 ! ( 26 - 13 ) !} (3)^{13} i^{13} + \frac{26 !}{14 ! ( 26 - 14 ) !} (3)^{12} i^{14} + \frac{26 !}{15 ! ( 26 - 15 ) !} (3)^{11} i^{15} + \frac{26 !}{16 ! ( 26 - 16 ) !} (3)^{10} i^{16} + \frac{26 !}{17 ! ( 26 - 17 ) !} (3)^{9} i^{17} + \frac{26 !}{18 ! ( 26 - 18 ) !} (3)^{8} i^{18} + \frac{26 !}{19 ! ( 26 - 19 ) !} (3)^{7} i^{19} + \frac{26 !}{20 ! ( 26 - 20 ) !} (3)^{6} i^{20} + \frac{26 !}{21 ! ( 26 - 21 ) !} (3)^{5} i^{21} + \frac{26 !}{22 ! ( 26 - 22 ) !} (3)^{4} i^{22} + \frac{26 !}{23 ! ( 26 - 23 ) !} (3)^{3} i^{23} + \frac{26 !}{24 ! ( 26 - 24 ) !} (3)^{2} i^{24} + \frac{26 !}{25 ! ( 26 - 25 ) !} (3)^{1} i^{25} + \frac{26 !}{26 ! ( 26 - 26 ) !} (3)^{0} i^{26}

Simplificar

\frac{26 !}{0 ! ( 26 - 0 ) !} (3)^{26} i^{0} : 1594323

Simplificar

\frac{26 !}{1 ! ( 26 - 1 ) !} (3)^{25} i^{1} : 138174663 i

Simplificar

\frac{26 !}{2 ! ( 26 - 2 ) !} (3)^{24} i^{2} : 172718325 i^{2}

Simplificar

\frac{26 !}{3 ! ( 26 - 3 ) !} (3)^{23} i^{3} : 4605822003 i^{3}

Simplificar

\frac{26 !}{4 ! ( 26 - 4 ) !} (3)^{22} i^{4} : 2648347650 i^{4}

Simplificar

\frac{26 !}{5 ! ( 26 - 5 ) !} (3)^{21} i^{5} : 38842432203 i^{5}

Simplificar

\frac{26 !}{6 ! ( 26 - 6 ) !} (3)^{20} i^{6} : 13594851270 i^{6}

Simplificar

\frac{26 !}{7 ! ( 26 - 7 ) !} (3)^{19} i^{7} : 129474774003 i^{7}

Simplificar

\frac{26 !}{8 ! ( 26 - 8 ) !} (3)^{18} i^{8} : 30750258825 i^{8}

Simplificar

\frac{26 !}{9 ! ( 26 - 9 ) !} (3)^{17} i^{9} : 205001725503 i^{9}

Simplificar

\frac{26 !}{10 ! ( 26 - 10 ) !} (3)^{16} i^{10} : 34850293335 i^{10}

Simplificar

\frac{26 !}{11 ! ( 26 - 11 ) !} (3)^{15} i^{11} : 168971119203 i^{11}

Simplificar

\frac{26 !}{12 ! ( 26 - 12 ) !} (3)^{14} i^{12} : \frac{1.01172 E 19 i ^{12}}{479001600}

Simplificar

\frac{26 !}{13 ! ( 26 - 13 ) !} (3)^{13} i^{13} : \frac{3 \cdot 4.72135 E 19 i ^{13}}{6227020800}

Simplificar

\frac{26 !}{14 ! ( 26 - 14 ) !} (3)^{12} i^{14} : 7040463300 i^{14}

Simplificar

\frac{26 !}{15 ! ( 26 - 15 ) !} (3)^{11} i^{15} : 18774568803 i^{15}

Simplificar

\frac{26 !}{16 ! ( 26 - 16 ) !} (3)^{10} i^{16} : 1290751605 i^{16}

Simplificar

\frac{26 !}{17 ! ( 26 - 17 ) !} (3)^{9} i^{17} : 2530885503 i^{17}

Simplificar

\frac{26 !}{18 ! ( 26 - 18 ) !} (3)^{8} i^{18} : 126544275 i^{18}

Simplificar

\frac{26 !}{19 ! ( 26 - 19 ) !} (3)^{7} i^{19} : 177606003 i^{19}

Simplificar

\frac{26 !}{20 ! ( 26 - 20 ) !} (3)^{6} i^{20} : 6216210 i^{20}

Simplificar

\frac{26 !}{21 ! ( 26 - 21 ) !} (3)^{5} i^{21} : 5920203 i^{21}

Simplificar

\frac{26 !}{22 ! ( 26 - 22 ) !} (3)^{4} i^{22} : 134550 i^{22}

Simplificar

\frac{26 !}{23 ! ( 26 - 23 ) !} (3)^{3} i^{23} : 78003 i^{23}

Simplificar

\frac{26 !}{24 ! ( 26 - 24 ) !} (3)^{2} i^{24} : 975 i^{24}

Simplificar

\frac{26 !}{25 ! ( 26 - 25 ) !} (3)^{1} i^{25} : 263 i^{25}

Simplificar

\frac{26 !}{26 ! ( 26 - 26 ) !} (3)^{0} i^{26} : i^{26}

= 1594323 + 138174663 i + 172718325 i^{2} + 4605822003 i^{3} + 2648347650 i^{4} + 38842432203 i^{5} + 13594851270 i^{6} + 129474774003 i^{7} + 30750258825 i^{8} + 205001725503 i^{9} + 34850293335 i^{10} + 168971119203 i^{11} + \frac{1.01172 E 19 i ^{12}}{479001600} + \frac{3 \cdot 4.72135 E 19 i ^{13}}{6227020800} + 7040463300 i^{14} + 18774568803 i^{15} + 1290751605 i^{16} + 2530885503 i^{17} + 126544275 i^{18} + 177606003 i^{19} + 6216210 i^{20} + 5920203 i^{21} + 134550 i^{22} + 78003 i^{23} + 975 i^{24} + 263 i^{25} + i^{26}

Simplificar

1594323 + 138174663 i + 172718325 i^{2} + 4605822003 i^{3} + 2648347650 i^{4} + 38842432203 i^{5} + 13594851270 i^{6} + 129474774003 i^{7} + 30750258825 i^{8} + 205001725503 i^{9} + 34850293335 i^{10} + 168971119203 i^{11} + \frac{1.01172 E 19 i ^{12}}{479001600} + \frac{3 \cdot 4.72135 E 19 i ^{13}}{6227020800} + 7040463300 i^{14} + 18774568803 i^{15} + 1290751605 i^{16} + 2530885503 i^{17} + 126544275 i^{18} + 177606003 i^{19} + 6216210 i^{20} + 5920203 i^{21} + 134550 i^{22} + 78003 i^{23} + 975 i^{24} + 263 i^{25} + i^{26} : 335544323 i + 33554432

= 335544323 i + 33554432

Escribe en forma compleja estándar:

33554432 + 335544323 i

= 33554432 + 335544323 i

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