1. Subtraction: You would subtract 25 from 500 to find out how many beads Niel had after making the necklaces but before buying additional beads.
2. Division: You would then divide the resulting number by 8 to find out how many beads he originally had per necklace (since he made 8 identical necklaces).
Therefore, the correct operations needed to solve this problem are:
subtraction and division