To solve the problem $e^{-5.8}$ and round it to four decimal places, let's proceed with the following steps:

1. Identify the Expression: The expression is $e^{-5.8}$, where $e$ is approximately equal to 2.71828.

2. Calculate: Using a calculator, we can find the value of $e^{-5.8}$.

   $e^{-5.8} \approx 0.0027$

3. Round: The answer to four decimal places is 0.0027.

So, the final answer is: $\boxed{0.0027}$

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:

1. What does the value of $e$ represent in mathematics?
2. How does the negative exponent in $e^{-5.8}$ affect the result?
3. How can you calculate expressions with other irrational constants, such as $\pi$ or $\sqrt{2}$?
4. What is the general rule for handling exponents with a base $e$?
5. Why is rounding to four decimal places sometimes required in calculations?

Tip: When working with exponential functions, remember that a negative exponent implies taking the reciprocal of the base raised to the positive value of that exponent.