\(x=\sqrt[3]{a^3+a+\dfrac{1}{3}\sqrt{27a^4+6a^2+\dfrac{1}{3}}}+\sqrt[3]{a^3+a-\dfrac{1}{3}\sqrt{27a^4+6a^2+\dfrac{1}{3}}}\)
\(=\sqrt[3]{a^3+a+\dfrac{1}{3}\cdot\sqrt{\left(3\sqrt{3}a^2\right)^2+2\cdot3\sqrt{3}\cdot a^2\cdot\dfrac{1}{\sqrt{3}}+\left(\dfrac{1}{\sqrt{3}}\right)^2}}+\sqrt[3]{a^3+a-\dfrac{1}{3}\cdot\sqrt{\left(3\sqrt{3}a^2\right)^2+2\cdot3\sqrt{3}\cdot a^2\cdot\dfrac{1}{\sqrt{3}}+\left(\dfrac{1}{\sqrt{3}}\right)^2}}\)
\(=\sqrt[3]{a^3+a+\dfrac{1}{3}\sqrt{\left(3\sqrt{3}a^2+\dfrac{1}{\sqrt{3}}\right)^2}}+\sqrt[3]{a^3+a-\dfrac{1}{3}\sqrt{\left(3\sqrt{3}a^2+\dfrac{1}{\sqrt{3}}\right)^2}}\)
\(=\sqrt[3]{a^3+a+\dfrac{1}{3}\left(3\sqrt{3}a^2+\dfrac{1}{\sqrt{3}}\right)}+\sqrt[3]{a^3+a-\dfrac{1}{3}\left(3\sqrt{3}a^2+\dfrac{1}{\sqrt{3}}\right)}\)
\(=\sqrt[3]{a^3+a+\sqrt{3}a^2+\dfrac{1}{3\sqrt{3}}}+\sqrt[3]{a^3+a-\sqrt{3}a^2-\dfrac{1}{3\sqrt{3}}}\)
\(=\sqrt[3]{a^3+3\cdot a^2\cdot\dfrac{1}{\sqrt{3}}+3\cdot a\cdot\dfrac{1}{3}+\left(\dfrac{1}{\sqrt{3}}\right)^3}+\sqrt[3]{a^3-3\cdot a^2\cdot\dfrac{1}{\sqrt{3}}+3\cdot a\cdot\dfrac{1}{3}-\left(\dfrac{1}{\sqrt{3}}\right)^3}\)
\(=\sqrt[3]{\left(a+\dfrac{1}{\sqrt{3}}\right)^3}+\sqrt[3]{\left(a-\dfrac{1}{\sqrt{3}}\right)^3}\)
\(=a+\dfrac{1}{\sqrt{3}}+a-\dfrac{1}{\sqrt{3}}=2a\)
