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

Step 1: Apply the Product Rule for Exponents

The product rule for exponents states that:

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

Using this rule:

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

Step 2: Simplify the Exponent

Simplify the exponent by adding:

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

Final Answer

The simplified expression is:

$\boxed{5^{-18}}$

Would you like any additional explanations or details?

Here are 5 related questions:

1. How would you simplify $5^3 \times 5^4$?
2. What is the value of $5^{-18}$ as a decimal?
3. How do you simplify $\frac{5^{-8}}{5^{-10}}$?
4. What is the general rule for multiplying powers with the same base?
5. How do negative exponents affect the value of an expression?

Tip: When multiplying two powers with the same base, add the exponents.