Is there any mathematical reason for this "digit-repetition-show"? The number $$\sqrt{308642}$$ has a crazy decimal representation : $$555.5555777777773333333511111102222222719999970133335210666544640008\cdots $$

Is there any mathematical reason for so many repetitions of the digits ?

A long block containing only a single digit would be easier to understand. This could mean that there are extremely good rational approximations. But here we have many long one-digit-blocks , some consecutive, some interrupted by a few digits. I did not calculate the probability of such a "digit-repitition-show", but I think it is extremely small.
Does anyone have an explanation ?
 A: The architect's answer, while explaining the absolutely crucial fact that $$\sqrt{308642}\approx 5000/9=555.555\ldots,$$ didn't quite make it clear why we get several runs of repeating decimals. I try to shed additional light to that using a different tool.
I want to emphasize the role of the binomial series. In particular the Taylor expansion
$$
\sqrt{1+x}=1+\frac x2-\frac{x^2}8+\frac{x^3}{16}-\frac{5x^4}{128}+\frac{7x^5}{256}-\frac{21x^6}{1024}+\cdots
$$
If we plug in $x=2/(5000)^2=8\cdot10^{-8}$, we get
$$
M:=\sqrt{1+8\cdot10^{-8}}=1+4\cdot10^{-8}-8\cdot10^{-16}+32\cdot10^{-24}-160\cdot10^{-32}+\cdots.
$$
Therefore
$$
\begin{aligned}
\sqrt{308462}&=\frac{5000}9M=\frac{5000}9+\frac{20000}9\cdot10^{-8}-\frac{40000}9\cdot10^{-16}+\frac{160000}9\cdot10^{-24}+\cdots\\
&=\frac{5}9\cdot10^3+\frac29\cdot10^{-4}-\frac49\cdot10^{-12}+\frac{16}9\cdot10^{-20}+\cdots.
\end{aligned}
$$
This explains both the runs, their starting points, as well as the origin and location of those extra digits not part of any run. For example, the run of $5+2=7$s begins when the first two terms of the above series are "active". When the third term joins in, we need to subtract a $4$ and a run of $3$s ensues et cetera.
A: Repeated same numbers in a decimal representation can be converted to repeated zeros by multiplication with $9$. (try it out)
so if we multiply $9 \sqrt{308642} =  \sqrt{308642 \times 81} = \sqrt{25 000 002}$  since this number is allmost $5000^2$ it has a lot of zeros in its decimal expansion 
