Estimate the value of (?) without finding the exact number: [(7.99)^2 − (13.001)^2 + (4.01)^3]^2 = ? Choose the closest approximate value.

Introduction / Context:
This question tests approximation by rounding to nearby convenient integers and using basic square/cube values. The given decimals are extremely close to 8, 13, and 4, so the inner expression can be estimated quickly and then squared to get a final approximation.

Given Data / Assumptions:

• Expression: [(7.99)^2 − (13.001)^2 + (4.01)^3]^2 • Use rounding: 7.99 ≈ 8, 13.001 ≈ 13, 4.01 ≈ 4 • Exact value not required; choose closest option

Concept / Approach:
Approximate each component using nearby integers: (7.99)^2 ≈ 8^2 = 64 (13.001)^2 ≈ 13^2 = 169 (4.01)^3 ≈ 4^3 = 64 Then combine: 64 − 169 + 64 ≈ −41, and square it: (−41)^2 = 1681. The closest listed value to 1681 is 1660.

Step-by-Step Solution:
1) Round terms: 7.99 ≈ 8, 13.001 ≈ 13, 4.01 ≈ 4 2) Compute approximate powers: 8^2 = 64, 13^2 = 169, 4^3 = 64 3) Combine inside brackets: 64 − 169 + 64 = 128 − 169 = −41 4) Square the result: (−41)^2 = 1681 5) Compare to options: 1660 is the nearest to 1681.

Verification / Alternative check:
Since the final expression is a square, it must be non-negative. That immediately eliminates negative options like −1660 and −1800. Also, the rounded estimate 1681 is much closer to 1660 than 1450 or 1800, so 1660 is the best approximation.

Why Other Options Are Wrong:
• Negative values: impossible because the entire expression is squared. • 1450: too far below 1681. • 1800: farther away than 1660.

Common Pitfalls:
• Forgetting the outer square and choosing a negative option. • Rounding 13.001 to 14 by mistake (wrong nearest integer).

Final Answer:
1660