2. Solve the equation

$$
2^{\sqrt[12]{x}}+2^{\sqrt[4]{x}}=2 \cdot 2^{\sqrt[6]{x}}
$$
Solution. Let's introduce the substitution $x=t^{12}, t \geq 0$, so $\sqrt[12]{x}=t, \sqrt[4]{x}=t^{3}$ and $\sqrt[6]{x}=t^{2}$. The original equation can be written in the form

$$
2^{t}+2^{t^{3}}=2 \cdot 2^{t^{2}}
$$

Since $2^{t}$ and $2^{t^{3}}, t$ and $t^{3}$ are non-negative real numbers, from the inequality between the arithmetic and geometric means, we get

$$
2^{t}+2^{t^{3}} \geq 2 \sqrt{2^{t} \cdot 2^{t^{3}}}=2 \cdot 2^{\frac{t+t^{3}}{2}} \geq 2 \cdot 2^{t^{2}}
$$

Equality holds if and only if $2^{t}=2^{t^{3}}$ and $t=t^{3}$. In both cases, we get the solutions $t_{1}=0, t_{2}=1, t_{3}=-1$. Clearly, $t_{3}$ cannot be a solution.

Thus, the solutions to the original equation are $x_{1}=0$ and $x_{2}=1$.