Without using a calculator (and without finding exact square-root values), estimate the number that should replace the question mark (?) in the expression: (√80.997 − √25.001) × (√120.98 + √16.02) Choose the closest approximate value.

Introduction / Context:
This question tests smart approximation with square roots by replacing awkward decimals with nearby perfect squares, so the expression becomes easy to evaluate mentally.

Given Data / Assumptions:

• Expression: (√80.997 − √25.001) × (√120.98 + √16.02) • Use nearby perfect squares for quick estimation • No exact root calculation is required

Concept / Approach:
When numbers are extremely close to perfect squares, we approximate: √(80.997) ≈ √81, √(25.001) ≈ √25, √(120.98) ≈ √121, √(16.02) ≈ √16. This works well because a tiny change in the number causes only a tiny change in the square root.

Step-by-Step Solution:
1) √80.997 ≈ √81 = 9 2) √25.001 ≈ √25 = 5 3) First bracket ≈ 9 − 5 = 4 4) √120.98 ≈ √121 = 11 5) √16.02 ≈ √16 = 4 6) Second bracket ≈ 11 + 4 = 15 7) Product ≈ 4 * 15 = 60

Verification / Alternative check:
Each original number is extremely close to its perfect square (81, 25, 121, 16), so the true value will be very close to 60. No large deviation is expected.

Why Other Options Are Wrong:
• 50 or 40: too small compared to 4 * 15. • 30 or 20: far too small; would require one bracket to be much smaller than 4 or 15.

Common Pitfalls:
• Approximating √120.98 as 10 instead of 11 (wrong nearby square). • Ignoring that √25.001 is still very close to 5, not 6.

Final Answer:
60