Đặt \(\left\{ \begin{array}{l}u = {2^x} - 4\\v = {4^x} - 2\end{array} \right. \Rightarrow PT \Leftrightarrow {u^3} + {v^3} = {\left( {u + v} \right)^3} \Leftrightarrow {u^3} + {v^3} = {u^3} + {v^3} + 3uv\left( {u + v} \right) \Leftrightarrow uv\left( {u + v} \right) = 0\)
\( \Leftrightarrow \left[ \begin{array}{l}u = 0\\v = 0\\u + v = 0\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}{2^x} - 4 = 0\\{4^x} - 2 = 0\\{4^x} + {2^x} - 6 = 0\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = 2\\x = \frac{1}{2}\\x = 1\end{array} \right. \Rightarrow \left\{ \begin{array}{l}{x_1} = 2\\{x_2} = \frac{1}{2}\\{x_3} = 1\end{array} \right. \Rightarrow {x_1} + {x_2} + {x_3} = \frac{7}{2}.\)
