Question

Calcule:
n = ∛(7+5√2) + ∛(7-5√2) , sabendo que n∈Z

Answer 1

Resposta:

Explicação passo-a-passo:

Lembrando que

$(a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}=a^{3}+b^{3}+3ab(a+b)$

$n=\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}$

Passando 5 para dentro da raiz quadrada

$n=\sqrt[3]{7+\sqrt{25.2}}+\sqrt[3]{7-\sqrt{25.2}}\\\\n=\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}$

Elevando ambos os membros ao cubo

$n^{3}=(\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}})^{3}\\\\n^{3}=7+\sqrt{50}+7-\sqrt{50}+3.\sqrt[3]{7+\sqrt{50}}.\sqrt[3]{7-\sqrt{50}}.(\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}})\\\\n^{3}=14+3\sqrt[3]{(7+\sqrt{50})(7-\sqrt{50})}.n\\\\n^{3}=14+3n\sqrt[3]{49-50}\\\\n^{3}=14+3n.\sqrt[3]{-1}\\\\n^{3}=14+3n.(-1)\\\\n^{3}=14-3n\\\\n^{3}+3n-14=0$

Logo os candidatos inteiros a raiz são

$\pm1,\pm2,\pm7,\pm14$

Fazendo a verificação dos candidatos veremos que

$2^{3}+3.2-14=8+6-14=14-14=0$

logo n=2