Finn drops a ball straight down from a height of $8$ decimeters $(\text{dm})$. With each bounce, it rises to $\dfrac{2}{3}$ of the height of the previous bounce.
**When it hits the ground the $10^\text{th}$ time, what is the total distance, in decimeters, that the ball has traveled?**
*Round your final answer the nearest hundredth.*
[[☃ numeric-input 1]] $\text{dm}$