Perfect-square identification with digit test and quick checks: Evaluate √64009 exactly by recognizing nearby squares (250^2 to 255^2) and confirming the units and tens patterns. What is the square root?

Introduction / Context:
Square-root questions on perfect squares can be solved quickly by testing nearby squares and using last-digit and tens-place cues. Here we locate √64009 by bracketing between known squares around 250^2.

Given Data / Assumptions:

• Target number N = 64009
• We suspect N is close to 250^2 = 62500
• We will test 251^2, 252^2, 253^2, etc.

Concept / Approach:
For k near 250, compute (250 + d)^2 = 250^2 + 2*250*d + d^2 to step upward efficiently. Also, the last digit 9 suggests roots ending in 3 or 7 for odd squares ending with 9; among options, 253 is a candidate.

Step-by-Step Solution:
251^2 = 63001252^2 = 63504253^2 = 64009 (exact match)

Verification / Alternative check:
Use (250 + 3)^2 = 250^2 + 1500 + 9 = 62500 + 1500 + 9 = 64009, confirming the result.

Why Other Options Are Wrong:
352, 523, and 532 produce squares far from 64009; 245^2 = 60025, which is too small.

Common Pitfalls:
Guessing without bracketing; ignoring simple expansion (a + b)^2 = a^2 + 2ab + b^2 which speeds mental checks.

Final Answer:
253