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Popular Álgebra >

simplificar (-5+5i)^5

Solución

simplificar

(- 5 + 5 i)^{5}

Solución

12500 - 12500 i

Pasos de solución

(- 5 + 5 i)^{5}

Aplicar el teorema del binomio:

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

a = - 5, b = 5 i

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

Expandir sumatorio

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

Simplificar

\frac{5 !}{0 ! ( 5 - 0 ) !} (- 5)^{5} (5 i)^{0} : - 3125

Simplificar

\frac{5 !}{1 ! ( 5 - 1 ) !} (- 5)^{4} (5 i)^{1} : 15625 i

Simplificar

\frac{5 !}{2 ! ( 5 - 2 ) !} (- 5)^{3} (5 i)^{2} : - 31250 i^{2}

Simplificar

\frac{5 !}{3 ! ( 5 - 3 ) !} (- 5)^{2} (5 i)^{3} : 31250 i^{3}

Simplificar

\frac{5 !}{4 ! ( 5 - 4 ) !} (- 5)^{1} (5 i)^{4} : - 15625 i^{4}

Simplificar

\frac{5 !}{5 ! ( 5 - 5 ) !} (- 5)^{0} (5 i)^{5} : 3125 i^{5}

= - 3125 + 15625 i - 31250 i^{2} + 31250 i^{3} - 15625 i^{4} + 3125 i^{5}

Simplificar

- 3125 + 15625 i - 31250 i^{2} + 31250 i^{3} - 15625 i^{4} + 3125 i^{5} : - 12500 i + 12500

= - 12500 i + 12500

Escribe en forma compleja estándar:

12500 - 12500 i

= 12500 - 12500 i

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