Example 23. Solve the equation

$$
x \sqrt[3]{x^{2}}=(\sqrt{x})^{x}
$$
Solution. The domain of admissible values of the given equation is determined by the condition $x>0$. For positive $x$, equation (25) is equivalent to the equation

$$
10^{x^{2 / 3} \cdot \lg x}=10^{(x \lg x) / 2}
$$

i.e., the equation

$$
\left(x^{2 / 3}-x / 2\right) \lg x=0
$$

From this, it follows that for $x>0$, equation (25) is equivalent to the system of equations

$$
\lg x=0, \quad x^{2 / 3}=x / 2,
$$

solving which we find two roots of equation (25): $x_{1}=8, x_{2}=1$.