The problem requires calculating the value of the product:

$\prod_{i=1}^8 5^i$

This can be rewritten as:

$5^1 \cdot 5^2 \cdot 5^3 \cdot \ldots \cdot 5^8 = 5^{1+2+3+\ldots+8}$

The sum of the integers from $1$ to $8$ is calculated using the formula for the sum of the first $n$ natural numbers:

$\text{Sum} = \frac{n(n+1)}{2}$

For $n = 8$:

$\text{Sum} = \frac{8(8+1)}{2} = \frac{8 \cdot 9}{2} = 36$

Thus, the product simplifies to:

$5^{36}$

So, the value of the product is:

$\boxed{5^{36}}$

Let me know if you need further details or clarification! Here are some related questions for practice:

1. What is the value of $\prod_{i=1}^{10} 2^i$?
2. How can the formula for the sum of natural numbers be applied to $\prod_{i=1}^n x^i$?
3. Calculate $\prod_{i=1}^5 3^i$.
4. What is the simplified result of $\prod_{i=1}^n a^i$ for any general $n$?
5. How does the product notation relate to exponential properties?

Tip: Always look for patterns or formulas that simplify repetitive operations like sums or products!