Number System — Solve for the unknown value (?) given the proportion: (sqrt(?)) / 196 = 72 / 56. Compute the exact value of ? that makes the equality true (do not approximate square roots).

Introduction / Context:
This question checks comfort with proportional reasoning and square roots. Instead of expanding into large products, we manipulate the given ratio to isolate sqrt(?) and then square to recover ?. Careful simplification prevents arithmetic errors and makes the computation fast and exact.

Given Data / Assumptions:

• (sqrt(?)) / 196 = 72 / 56.
• All quantities are real and positive in the context of basic number system problems.
• No approximation is needed; the task asks for the exact value of ?.

Concept / Approach:
Use equivalence of ratios and simplify the right-hand fraction before solving. Once sqrt(?) is isolated, square both sides to remove the square root. Keep arithmetic exact to avoid rounding mistakes. Recognizing common factors in 72 and 56 simplifies the process immediately.

Step-by-Step Solution:
1) Simplify 72 / 56 by dividing numerator and denominator by 8: 72/56 = 9/7.2) Set up equality: (sqrt(?)) / 196 = 9 / 7.3) Multiply both sides by 196: sqrt(?) = 196 * (9/7) = 28 * 9 = 252.4) Square both sides to solve for ?: ? = 252^2.5) Compute 252^2: 252 × 252 = 63504.

Verification / Alternative check:
Back-substitute: sqrt(63504) = 252, then 252/196 = 9/7, which equals 72/56 after simplification. Everything is consistent, confirming 63504 is correct.

Why Other Options Are Wrong:
62504 and 63540 are near-miss digit transpositions; 252 confuses sqrt(?) with ?; 324 is 18^2 and unrelated to the derived 252.

Common Pitfalls:
Simplifying 72/56 incorrectly; forgetting to square after isolating sqrt(?); computing 252^2 with digit slips; treating 252 (the root) as the final answer instead of squaring back to ?.

Final Answer:
63504