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Popular Álgebra >

simplificar (5+2i)^6

Solución

simplificar

(5 + 2 i)^{6}

Solución

- 15939 + 18460 i

Pasos de solución

(5 + 2 i)^{6}

Aplicar el teorema del binomio:

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

a = 5, b = 2 i

= i = 0 \sum 6 (i 6) \cdot 5^{(6 - i)} (2 i)^{i}

Expandir sumatorio

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

Simplificar

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 5^{6} (2 i)^{0} : 15625

Simplificar

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 5^{5} (2 i)^{1} : 37500 i

Simplificar

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 5^{4} (2 i)^{2} : 37500 i^{2}

Simplificar

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 5^{3} (2 i)^{3} : 20000 i^{3}

Simplificar

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 5^{2} (2 i)^{4} : 6000 i^{4}

Simplificar

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 5^{1} (2 i)^{5} : 960 i^{5}

Simplificar

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 5^{0} (2 i)^{6} : 64 i^{6}

= 15625 + 37500 i + 37500 i^{2} + 20000 i^{3} + 6000 i^{4} + 960 i^{5} + 64 i^{6}

Simplificar

15625 + 37500 i + 37500 i^{2} + 20000 i^{3} + 6000 i^{4} + 960 i^{5} + 64 i^{6} : 18460 i - 15939

= 18460 i - 15939

Escribe en forma compleja estándar:

- 15939 + 18460 i

= - 15939 + 18460 i

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