A resposta correta é: Letra E (errado) Para responder este exercicio, temos que utilizar a formula de De Moivre Zn=rn(cosnO +isen(n)O) onde, O = teta como o exercicio nos da z³= α Teremos z³ = α³ (cos3O + isen3O)z³ = ³√α (cos(O/3) + i sen (O/3)) Se utilizarmos O = O +2Kπ, onde o nosso K = 0,1,2, teremos K=0 Z1 = ³√α (cos(O/3) + i sen (O/3)) k=1 Z2 = ³√α (cos(O+2π/3) + i sen (O+2π/3)) K=2 Z3 = ³√α (cos(O+4π/3) + i sen (O+4π/3)) Isto nos mostra que no plano complexo temos pontos igualmente espaçados com centro na origem, formando um triângulo equilatero

Para algum número real α não nulo, na representação
geométrica das soluções complexas z1, z2 e z3 da equação
z3 = α, z1, z2 e z3 podem ser vértices de um triângulo retângulo.

Julgue os itens seguintes, a respeito de números complexos e
funções de variáveis complexas.


Gabarito: Certo. 							<img src="data:image/png;base64,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">&nbsp; &nbsp;Fazendo: a = 4, b = 9 e c = 25. 							<img src="data:image/png;base64,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"> Se os números forem zero ou 1 é possível ter uma raiz igual mas nunca maior que a soma das raízes em separados.

Se a, b e c forem números reais não negativos,
então <img src='https://s3.amazonaws.com/mapastatic/5f4e3e840905e967a103146f/imgs_cespe-2017-professor-basico-conhecimentos-basicos1-prova/90.jpg'/>.

A respeito de números reais e números complexos, julgue os itens
subsecutivos.


Se a, b e c forem números reais não negativos,
então <img alt='Imagem questão' src='https://d28sq8dmsjgyhi.cloudfront.net/5f4e3e840905e967a103146f/imgs_cespe-2017-professor-basico-conhecimentos-basicos1-prova/90.jpg'/>.

Gabarito: Certo.&nbsp;Raízes cúbicas do número complexo 1&nbsp;Passo 1: Transformar 1 para sua forma trigonométrica.&nbsp;z = |z| * (cosα + i senα), temos z = 1 * (cos0° + i seno0º)&nbsp;&nbsp;Passo 2: Calcular os módulos das raízes ||z|| = Raiz de n de |z|, logo temos raiz cúbica de 1 igual a 1.&nbsp;&nbsp;Passo 3: Calcular o αk de cada raiz pela fórmula αk = (α + 2kπ)/n&nbsp;Temos α0 = 0, α1= 2π/3 e α2=4π/3&nbsp;&nbsp;Passo 4: Calcular as raízes pela fórmula de moivre R = ||z|| * (cosαk + i senαk)&nbsp;R1= 1 * (cos 0 + isen 0) = 1 + 0i ou 1&nbsp;R2 = 1 * (cos2π/3 + i sen2π/3) = cos 120º + isen120º = -1/2 + √3i/2&nbsp;R3 = 1 * (cos4π/3 + i sen 4π/3) = cos 240º + isen 240º = -1/2 - √3i/2&nbsp;&nbsp;Passo 5: somar as raízes para confirmar se é real.&nbsp;Assim, R1 + R2 + R3 = 0.

Se z1, z2 e z3 forem as raízes cúbicas complexas de 1, então
o número z1 + z2 + z3 será real.

Gabarito: Errado.Temos que: i4 = i2 * i² = (-1) * (-1) = 1.&nbsp;Logo, a parte imaginária de z4 é diferente de zero.

Se a parte imaginária de z for diferente de zero, então a parte
imaginária de z4 também será diferente de zero.

Questões de Matemática - Números complexos