2. Find all triples of positive real numbers $(x, y, z)$ which satisfy the system
$$
\left\{\begin{array}{l}
\sqrt[3]{x}-\sqrt[3]{y}-\sqrt[3]{z}=64 \\
\sqrt[4]{x}-\sqrt[4]{y}-\sqrt[4]{z}=32 \\
\sqrt[6]{x}-\sqrt[6]{y}-\sqrt[6]{z}=8
\end{array}\right.
$$
Answer: no solution
Solution: Using the first and the third equations, we find that
$$
(8+\sqrt[6]{y}+\sqrt[6]{z})^{2}=64+\sqrt[3]{y}+\sqrt[3]{z}
$$

Simplifying, we obtain the following equation
$$
8 \sqrt[6]{y}+8 \sqrt[6]{z}+\sqrt[6]{y z}=0,
$$
which has no solution.