Raíz cúbica da unidade.

por Bergamotinha OwO Sex 13 maio 2022, 22:08

Se w ≠ 1 e é raíz cúbica da unidade, quanto vale (1 + w - w²)^7 ?

a) 128w
b) 128w²
c) -128w
d) -128w²

Resp.: Sem gabarito.

Boa noite pessoal!
Nessa questão eu achei primeiro as raízes cúbicas, chegando a 2 opções para o w e trabalhei em cima delas.
Porém, eu n acho resposta... Apenas consigo achar a parte de (+128), ficando entre a e b.

Vlww! cheers

por **Elcioschin** Sex 13 maio 2022, 22:26

Então mostre o passo-a-passo dose cálculos para podermos conferir.
As duas raízes válidas são w' = - 1/2 + i.√3/2 e w" = - 1/2 - i.√3/2

por Bergamotinha OwO Sex 13 maio 2022, 22:43

Essas são exatamente as raízes que achei!
Dps de ter descoberto elas, eu abri em duas situações:
1. Considerando w = cis(2π/3)
2. Considerando w = cis(4π/3)

Após isso eu substitui ambos os valores de w na expressão dada.
Porém, nessa substituição, não encontro nenhuma das opções, apenas a parte do módulo, que seria 2^7 dando (+128)

por **Elcioschin** Sáb 14 maio 2022, 15:19

Então mostre o passo-a-passo até onde vc chegou para podermos conferir.

por **tales amaral** Sáb 14 maio 2022, 16:51

[latex]\begin{align*} 1 + w - w^2 &= 1+w+w^2-2w^2 \\~\\ &= \dfrac{w^3-1}{w-1} -2w^2\\~\\ &= -2w^2 \end{align*}[/latex]

[latex]\left(-2w^2 \right )^7 = -128w^{14} = -128w^2 [/latex]

por Bergamotinha OwO Dom 15 maio 2022, 16:09

Fiz dessa maneira:
-----------------------------------

Raíz cúbica da unidade. Svg+xml;base64,<?xml version='1.0' encoding='UTF-8'?>
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Apliquei a segunda lei de De moivre, com 0 ≤ k < 3
Nisso, w pode ser 3 valores:
1. k = 0 ----> w = 1
2. k = 1 ----> w = (-1/2) + (i√3)/2 ----> w² = cis(4π/3)
3. k = 2 ----> w = (-1/2) - (i√3)/2 ----> w² = cis(2π/3)

Com isso, eu abri em dois casos:
1. Utilizando o 2º valor de w.

= (1 + i√3)^7 = [cis(π/3)]^7 = cis(2π/3)

2. Utilizando o 3º valor de w.

= (1 - i√3)^7 = [cis(5π/3)]^7 = cis(5π/3)

Fazendo desse jeito eu não acho os resultados....