9. The number of real roots of the equation $\left(\frac{3}{19}\right)^{x}+\left(\frac{5}{19}\right)^{2}+\left(\frac{11}{19}\right)^{x}=2 \sqrt{x-1}$ is
9. 1.

From $x-1 \geqslant 0$, we get $x \geqslant 1$. Since $f(x)=\left(\frac{3}{19}\right)^{x}+\left(\frac{5}{19}\right)^{x}+\left(\frac{11}{19}\right)^{x}$ is a decreasing function on $[1,+\infty)$, when $x=1$, $f_{\text {max }}=1$. Also, $g(x)=2 \sqrt{x-1}$ is an increasing function on $[1,+\infty)$, and when $x=1$, $g_{\text {min }}=0$. On the same coordinate system, from the graphical analysis, it is easy to see that the graphs of the two functions intersect at only one point, meaning the original equation has a unique solution.