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PT
$\Leftrightarrow (\sqrt[3]{x-1} + \sqrt[3]{x+1})^3= (x\sqrt[3]{2})^3 $
$\Leftrightarrow x-1+x+1+3\sqrt[3]{x-1}.\sqrt[3]{x+1}(\underbrace{\sqrt[3]{x-1} + \sqrt[3]{x+1}}_{x\sqrt[3]{2}})= 2x^3 $
$\Leftrightarrow 2x+3\sqrt[3]{x^2-1}.x\sqrt[3]{2}= 2x^3 $
$\Leftrightarrow 2x+3x\sqrt[3]{2x^2-2}= 2x^3 $
$\Leftrightarrow \left[ {\begin{matrix} x=0\\ 2+3\sqrt[3]{2x^2-2}= 2x^2\end{matrix}} \right. $
$\Leftrightarrow \left[ {\begin{matrix} x=0\\ 3\sqrt[3]{2x^2-2}= 2x^2-2\end{matrix}} \right. $
$\Leftrightarrow \left[ {\begin{matrix} x=0\\ 3(2x^2-2)= (2x^2-2)^3\end{matrix}} \right. $
$\Leftrightarrow \left[ {\begin{matrix} x=0\\ 2x^2-2=0\\3= (2x^2-2)^2\end{matrix}} \right. $
$\Leftrightarrow \left[ {\begin{matrix} x=0\\ x=\pm 1\\\pm \sqrt 3= 2x^2-2\end{matrix}} \right. $
$\Leftrightarrow \left[ {\begin{matrix} x=0\\ x=\pm 1\\2x^2=2\pm \sqrt 3\end{matrix}} \right. $
$\Leftrightarrow \left[ {\begin{matrix} x=0\\ x=\pm 1\\x=\pm\sqrt {\dfrac{2\pm \sqrt 3}{2}}\end{matrix}} \right. $
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