To round the number $2.76179999427$ to 5 decimal places:

1. Identify the first 5 decimal places: $2.76179$
2. Look at the 6th decimal place: $9$
3. Since the 6th decimal place is $9$, which is greater than or equal to 5, round up the 5th decimal place.

Thus, $2.76179999427$ rounded to 5 decimal places is $\mathbf{2.76180}$.

Would you like more details or have any questions? Here are some related questions:

1. How do you round a number to a specific decimal place?
2. What is the difference between truncating and rounding a number?
3. How do you round numbers when working with significant figures?
4. Can you provide an example of rounding with negative decimal places?
5. How does rounding affect the precision of a number?

Tip: Always check the digit immediately after the place you are rounding to determine whether to round up or down.