To arrange the given functions from the largest to the smallest rate of growth, we need to analyze each function's asymptotic behavior as $n \to \infty$.

1. $(5n)!$ (Factorial): Factorial growth is extremely fast. The growth of $(5n)!$ is much faster than any polynomial or exponential function, including $5^n$ or $n^5$.

2. $5^n$ (Exponential growth): Exponential growth (base $5$) grows faster than any polynomial function (such as $n^5$) or logarithmic growth, but slower than factorial growth.

3. $n^5$ (Polynomial): Polynomial functions grow slower than exponential and factorial functions but faster than logarithmic or root functions.

4. $n^5 / 500$ (Polynomial, divided by a constant): This is just a constant multiple of $n^5$, so it grows at the same rate as $n^5$, but it's scaled down by a factor of $1/500$, which does not change its growth rate.

5. $n \log n$ (Log-Linear): This grows faster than logarithmic functions but slower than polynomials.

6. $\log n / 500$ (Logarithmic, divided by a constant): Logarithmic growth is slow, and scaling by a constant factor does not change the growth order.

7. $5 \log n$ (Logarithmic, multiplied by a constant): Like $\log n$, logarithmic growth is slow, and multiplying by a constant does not change its growth rate.

8. $\sqrt{n}$ (Root function): Root functions grow slower than polynomial functions and logarithmic functions (in the long run), but they are faster than constant functions.

Final Order (from largest to smallest growth):

1. $(5n)!$ (Factorial growth)
2. $5^n$ (Exponential growth)
3. $n^5$ (Polynomial growth)
4. $n^5 / 500$ (Polynomial growth, scaled down)
5. $n \log n$ (Log-linear growth)
6. $5 \log n$ (Logarithmic growth)
7. $\log n / 500$ (Logarithmic growth)
8. $\sqrt{n}$ (Root function)

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Would you like more details on any specific function or growth rates?

Here are some related questions:

1. What are the asymptotic notations like Big O for each of these functions?
2. How does the growth rate of factorial compare with other types of functions like exponential?
3. What is the importance of logarithmic growth in algorithm complexity analysis?
4. How does exponential growth affect performance in large-scale computations?
5. How do we analyze the space complexity of algorithms with these growth rates?

Tip: Understanding the general behavior of these functions at large $n$ can help you predict the performance of algorithms in real-world scenarios.