9. (This question is worth 16 points) It is known that $\left\{a_{n}\right\}$ is a geometric sequence with all terms being positive, and $a_{50}, a_{51}$ are the two different solutions of the equation
$$
100 \lg ^{2} x=\lg (100 x)
$$

Find the value of $a_{1} a_{2} \cdots a_{100}$.
For $k=50,51$, we have $100 \lg ^{2} a_{k}=\lg \left(100 a_{k}\right)=2+\lg a_{k}$, which means
$$
100\left(\lg a_{k}\right)^{2}-\lg a_{k}-2=0 \text {. }
$$

Therefore, $\lg a_{50}, \lg a_{51}$ are the two distinct real roots of the quadratic equation $100 t^{2}-t-2=0$, hence $\lg \left(a_{50} a_{51}\right)=\lg a_{50}+\lg a_{51}=\frac{1}{100}$, which means $a_{50} a_{51}=10^{\frac{1}{100}}$. .8 points By the properties of geometric sequences, we know that $a_{1} a_{2} \cdots a_{100}=\left(a_{50} a_{51}\right)^{50}=\left(10^{\frac{1}{100}}\right)^{50}=\sqrt{10}$.
16 points