48. (SWE 1) Determine all positive roots of the equation $x^{x}=1 / \sqrt{2}$.
48. Put $f(x)=x \ln x$. The given equation is equivalent to $f(x)=f(1 / 2)$, which has the solutions $x_{1}=1 / 2$ and $x_{2}=1 / 4$. Since the function $f$ is decreasing on $(0,1 / e)$, and increasing on $(1 / e,+\infty)$, this equation has no other solutions.