Question

(Colégio Naval-82)
$\sqrt[3]{10+6\sqrt{3} }$ é igual a:
$a) 1+\sqrt{7} \\b)1+\sqrt{6} \\c)1+\sqrt{5}\\d)1+\sqrt{3}\\e) 1+\sqrt{2}$

Answer 1

Explicação passo-a-passo:

$\sf \sqrt[3]{10+6\sqrt{3}}$

$\sf =\sqrt[3]{4+6+2\sqrt{3}+4\sqrt{3}}$

$\sf =\sqrt[3]{4+2\cdot3+2\sqrt{3}+4\sqrt{3}}$

$\sf =\sqrt[3]{4+2\sqrt{3}\cdot\sqrt{3}+2\sqrt{3}+4\sqrt{3}}$

$\sf =\sqrt[3]{4+4\sqrt{3}+2\sqrt{3}+2\sqrt{3}\cdot\sqrt{3}}$

$\sf =\sqrt[3]{4\cdot(1+\sqrt{3})+2\sqrt{3}\cdot(1+\sqrt{3})}$

$\sf =\sqrt[3]{(1+\sqrt{3})\cdot(4+2\sqrt{3})}$

$\sf =\sqrt[3]{(1+\sqrt{3})\cdot(1+3+\sqrt{3}+\sqrt{3})}$

$\sf =\sqrt[3]{(1+\sqrt{3})\cdot[1+(\sqrt{3})^2+\sqrt{3}+\sqrt{3}]}$

$\sf =\sqrt[3]{(1+\sqrt{3})\cdot[1+\sqrt{3}+\sqrt{3}+(\sqrt{3})^2]}$

$\sf =\sqrt[3]{(1+\sqrt{3})\cdot[1\cdot(1+\sqrt{3})+\sqrt{3}\cdot(1+\sqrt{3})]}$

$\sf =\sqrt[3]{(1+\sqrt{3})\cdot(1+\sqrt{3})\cdot(1+\sqrt{3})}$

$\sf =\sqrt[3]{(1+\sqrt{3})^3}$

$\sf =1+\sqrt{3}$

Letra D