Number System — Evaluate the square exactly: 1299 × 1299. Select the correct answer.

Introduction / Context:
Squaring a near-1000 or near-1300 number can be made efficient with a near-base approach. Here, 1299 is one less than 1300, so the binomial (a − b)^2 expansion is especially convenient and reduces risk of arithmetic errors.

Given Data / Assumptions:

• Compute 1299^2 exactly.
• Use (a − b)^2 = a^2 − 2ab + b^2 with a = 1300 and b = 1.
• Carefully handle large-place multipliers to avoid dropped zeros.

Concept / Approach:
Let 1299 = 1300 − 1. Then (1300 − 1)^2 = 1300^2 − 2*1300*1 + 1^2. This avoids full long multiplication while remaining exact and transparent.

Step-by-Step Solution:
1) 1300^2 = 1,690,000.2) 2*1300*1 = 2600; subtract: 1,690,000 − 2,600 = 1,687,400.3) Add 1^2 = 1 → 1,687,401.4) Therefore, 1299^2 = 1,687,401.

Verification / Alternative check:
Check last two digits: 99 × 99 ends with 01, matching the result's ending 01. The magnitude also aligns with being slightly less than 1300^2 (1,690,000), confirming plausibility.

Why Other Options Are Wrong:
1684701 and 1685401 are off by hundreds or thousands due to middle-term mistakes; 1692601 exceeds 1300^2; 1585301 is far too small, indicating a base or place-value error.

Common Pitfalls:
Forgetting to subtract 2ab; misplacing zeros in 1300^2; arithmetic slips when subtracting 2600 from 1,690,000.

Final Answer:
1687401