Respuestas

2012-06-25T23:41:50+02:00

vamos a resolver este ejercicio, paso a paso (solo es un ejemplo para que te guies)

(3 + 3i)^5


Donde 

n = 5



➊ Recordando que

1 – i = r [cos (α) + i sen (α) ]


r = √ [ a² + b²]


α = tan⁻¹ = (b/a)



(a + bi)ⁿ = [ r ] ⁿ [cos (α)n + i sen (α)n ]




➋ Resolvemos

r = √ [ a² + b²]


r = √ [ [3]² + [3]²]


r = √ [9 + 9]


r = √ [18]




➌ α = tan⁻¹ = (b/a)

α = tan⁻¹ = (3/3)

α = 45°




➍ Sustituimos datos

r [cos (α) + i sen (α) ]


√ [18] [cos (45°) + i sen (45°) ]




➎ Sustituimos datos y resolvemos

(a + bi)ⁿ = [ r ] ⁿ [cos (α)n + i sen (α)n ]


(3 + 3i)^5 = [ √ [18] ]^5 [cos (45°)5 + i sen (45°)5 ]


(3 + 3i)^5 = [ √ [18] ]^5 [cos (225°) + i sen (225°) ]







➏ Recordando

cos (225°) = - 0.7071

sen (225°) = - 0.7071


(3 + 3i)^5 = [ √ [18] ]^5 [ - 0.7071 – 0.7071i]



Esta es la respuesta
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(3 + 3i)^5 = - 972 – 972i
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