\(x=\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}\)
\(2x=\sqrt[3]{8\sqrt{5}+16}-\sqrt[3]{8\sqrt{5}-16}\)
\(=\sqrt[3]{1^3+3.1^2.\sqrt{5}+3.1.\left(\sqrt{5}\right)^2+\left(\sqrt{5}\right)^3}-\sqrt[3]{-1^3+3.1^2.\sqrt{5}-3.1.\left(\sqrt{5}\right)^2+\left(\sqrt{5}\right)^3}\)
\(=\sqrt[3]{\left(1+\sqrt{5}\right)^3}-\sqrt[3]{\left(\sqrt{5}-1\right)^3}\)
\(=1+\sqrt{5}-\left(\sqrt{5}-1\right)=2\)
Suy ra \(x=1\).
Cách khác:
\(a=\sqrt[3]{\sqrt{5}+2},b=\sqrt[3]{\sqrt{5}-2}\)
\(x^3=\left(a-b\right)^3=a^3-b^3-3ab\left(a+b\right)\)
\(=\sqrt{5}+2-\left(\sqrt{5}-2\right)-3x=4-3x\) (vì \(ab=\sqrt[3]{\left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right)}=\sqrt[3]{5-4}=1\))
\(\Leftrightarrow x^3+3x-4=0\)
\(\Leftrightarrow\left(x-1\right)\left(x^2+x+4\right)=0\)
\(\Leftrightarrow x=1\).
