To simplify $(5^{-8})(5^{-10})$ and rewrite it in the form $5^n$, we can use the property of exponents:

$a^m \cdot a^n = a^{m+n}$

In this case, we have:

$5^{-8} \cdot 5^{-10} = 5^{(-8) + (-10)}$

Simplifying the exponent:

$5^{(-8) + (-10)} = 5^{-18}$

So, the answer is:

$5^{-18}$

Would you like further clarification on this process?

Here are five additional related questions to expand your understanding:

1. How would you simplify $(5^{3})(5^{-4})$?
2. What is the general rule for multiplying two powers with the same base?
3. How would the answer change if the expression was $(5^{-8})(5^{10})$?
4. How can you express $\frac{5^{-8}}{5^{-10}}$ in terms of a single power of 5?
5. What happens if you multiply $(5^a)(5^b)$ where $a$ and $b$ are any real numbers?

Tip: When multiplying powers with the same base, add the exponents. This property is very useful for simplifying expressions quickly.