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 #1
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syllogist Jun 19, 2015
 #1
avatar+125 
+8

$${\mathtt{y}} = {{\mathtt{x}}}^{{\mathtt{3}}}$$

$${{\mathtt{y}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{y}} = {\mathtt{2}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{y}} = {\mathtt{2}}\\
{\mathtt{y}} = -{\mathtt{1}}\\
\end{array} \right\}$$

$${{\mathtt{x}}}^{{\mathtt{3}}} = {\mathtt{2}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\frac{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}{{\mathtt{2}}}}\\
{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,\small\textbf+\,}}{{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}{{\mathtt{2}}}}\\
{\mathtt{x}} = {{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\mathtt{0.629\: \!960\: \!524\: \!947\: \!436\: \!6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.091\: \!123\: \!635\: \!972\: \!428\: \!7}}{i}\\
{\mathtt{x}} = {\mathtt{\,-\,}}\left({\mathtt{0.629\: \!960\: \!524\: \!947\: \!436\: \!6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.091\: \!123\: \!635\: \!972\: \!428\: \!7}}{i}\right)\\
{\mathtt{x}} = {\mathtt{1.259\: \!921\: \!049\: \!894\: \!873\: \!2}}\\
\end{array} \right\}$$

$${{\mathtt{x}}}^{{\mathtt{3}}} = -{\mathtt{1}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\
{\mathtt{x}} = -{\mathtt{1}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}\left({\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.866\: \!025\: \!403\: \!785}}{i}\right)\\
{\mathtt{x}} = {\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.866\: \!025\: \!403\: \!785}}{i}\\
{\mathtt{x}} = -{\mathtt{1}}\\
\end{array} \right\}$$

So the real solutions are:

  • $${\mathtt{x}} = {{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}$$
  • $${\mathtt{x}} = -{\mathtt{1}}$$
syllogist Jun 19, 2015