1. **Stating the problem:** We need to identify which numbers among 153, 257, 408, and 441 are definitely not perfect squares without performing detailed calculations. 2. **Important rule:** A perfect square ends with certain digits in base 10. For example, perfect squares can only end with 0, 1, 4, 5, 6, or 9. Numbers ending with 2, 3, 7, or 8 cannot be perfect squares. 3. **Check the last digit of each number:** - 153 ends with 3 - 257 ends with 7 - 408 ends with 8 - 441 ends with 1 4. **Apply the rule:** - 153 ends with 3 → cannot be a perfect square - 257 ends with 7 → cannot be a perfect square - 408 ends with 8 → cannot be a perfect square - 441 ends with 1 → could be a perfect square 5. **Additional check for 441:** 441 is a known perfect square since $21^2 = 441$. **Final answer:** The numbers that are definitely not perfect squares are 153, 257, and 408.