2. 1 Detailed Explanation: It is clear that $x \geqslant 1$. The function $f(x)=\left(\frac{3}{17}\right)^{x}+\left(\frac{5}{17}\right)^{x}+\left(\frac{9}{17}\right)^{x}$ is a decreasing function on $[1,+\infty)$, so when $x=1$, $f(x)_{\max }=1$. Also, $g(x)=\sqrt{x-1}$ is an increasing function on $[1,+\infty)$, so when $x=1$, $g(x)_{\min }=0$. From the graph, it is known that the original equation has only one solution.
2. The number of real roots of the equation $\left(\frac{3}{17}\right)^{x}+\left(\frac{5}{17}\right)^{x}+\left(\frac{9}{17}\right)^{x}=\sqrt{x-1}$ is $\qquad$ .