Example 19. Solve the equation

$$
6 \sqrt[x]{9}-13 \sqrt[x]{6}+6 \sqrt[x]{4}=0
$$
Solution. The domain of admissible values of the given equation consists of all natural numbers greater than 1.

Equation (19) is homogeneous; dividing both sides of the equation by $\sqrt[x]{\overline{4}}$ and setting $t=\sqrt[x]{3 / 2}$, we obtain the equation

$$
6 t^{2}-13 t+6=0
$$

the roots of which are $t_{1}=3 / 2, t_{2}=2 / 3$.

Thus, equation (19) is equivalent to the system of equations

$$
\sqrt[x]{3 / 2}=3 / 2, \quad \sqrt[x]{3 / 2}=(3 / 2)^{-1}
$$

which have no solutions. Therefore, equation (19) also has no roots.

Remark. In some textbooks and guides, the domain of definition, for example, of the function $y=\sqrt[x]{2}$ is considered to be the set of all positive numbers (and sometimes all $x$ such that $x \neq 0$).

However, adhering to the precise definition of the root, it should be considered that the function $y=\sqrt[x]{2}$ is defined only for natural $x \geqslant 2$.

Therefore, the functions $y=2^{1 / x}$ and $y=\sqrt[x]{2}$ cannot be considered identical: they have different domains of definition.