To find the smallest and largest 8-digit numbers using the digits $7, 2, 1, 0, 4, 9, 8$, with repetition allowed, let's break it down:

Smallest Number:

• To get the smallest number, we arrange the digits in ascending order, making sure the first digit is not zero (as the first digit of a number can't be zero in this case).
• The smallest digit that is non-zero is 1, so we will use 1 as the first digit.
• After the first digit, we use the smallest digits (including zero) for the remaining places.

So, the smallest number we can form is: 10012478

Largest Number:

• To get the largest number, we arrange the digits in descending order.
• The largest digit available is 9, so we'll use the largest available digits for each place.

So, the largest number we can form is: 99877421

Final Answer:

• Smallest Number: 10012478
• Largest Number: 99877421

Would you like further details on this or have any questions?

Here are some additional related questions:

1. How many distinct 8-digit numbers can be formed using these digits?
2. What is the second smallest number you can form?
3. What is the second largest number you can form?
4. How would the answer change if repetition was not allowed?
5. Can you form an 8-digit number that is divisible by 9 using these digits?

Tip: Always ensure the first digit of a number is non-zero when forming numbers from a set of digits.