Question

Rút gọn \(\sqrt[3]{16-8\sqrt{5}}+\sqrt[3]{16+8\sqrt{5}}\)

Answer 1

\(x=\sqrt[3]{16-8\sqrt{5}}+\sqrt[3]{16-8\sqrt{5}}\)

\(\Rightarrow x^3=32+3\sqrt[3]{16^2-8^2.5}\left(\sqrt[3]{16-8\sqrt{5}}+\sqrt[3]{16+8\sqrt{5}}\right)\)

\(\Rightarrow x^3=32-12x\)

\(\Rightarrow x^3+12x-32=0\)

\(\Rightarrow\left(x-2\right)\left(x^2+2x+16\right)=0\)

\(\Rightarrow x=2\)

Vậy \(\sqrt[3]{16-8\sqrt{5}}+\sqrt[3]{16+8\sqrt{5}}=2\)

Answer 2

$x=\sqrt[3]{316-8\sqrt{5}}+\sqrt[3]{316-8\sqrt{5}}$

$\Rightarrow x^3=32+3\sqrt[3]{3{16}^2-8^2.5}\left(\sqrt[3]{316-8\sqrt{5}}+\sqrt[3]{316+8\sqrt{5}}\right)$

$\Rightarrow x^3=32-12x$

$\Rightarrow x^3+12x-32=0$

$\Rightarrow \left(x-2\right)\left(x^2+2x+16\right)=0$

$\Rightarrow x=2$

Vậy $\sqrt[3]{316-8\sqrt{5}}+\sqrt[3]{316+8\sqrt{5}}=2$