Comparison of x and y from two statements (use the mapping below): I. x^2 = 729 II. y = √729 (principal square root) Use this mapping for the options: 1 → x > y, 2 → x < y, 3 → x = y, 4 → Relationship cannot be determined from the information given.

Introduction / Context:
This comparison problem provides two separate statements about x and y. You must deduce the relationship between x and y without assuming anything beyond standard definitions. The twist is that I gives a squared equation with two solutions, while II fixes y via a principal square root, which is always the non-negative value.

Given Data / Assumptions:

• I: x^2 = 729
• II: y = √729, the principal square root
• Principal square root √n is defined as the unique non-negative number whose square is n.

Concept / Approach:
From I, solve for x by taking square roots, remembering both positive and negative roots. From II, determine y directly using the principal root. Then test all possible values of x against y to see if a single strict relation (>, <, =) always holds. If different cases give different relations, the comparison is indeterminate.

Step-by-Step Solution:
From I: x^2 = 729 ⇒ x = ±27From II: y = √729 = 27 (principal root)Case 1: x = 27 ⇒ x = yCase 2: x = −27 ⇒ x < y

Verification / Alternative check:
The set of solutions to x^2 = 729 is exactly {−27, 27}. Since both are compatible with the data, and these two cases produce different relations (equality in one, “less than” in the other), there is no unique comparison that always holds.

Why Other Options Are Wrong:
“x > y”, “x < y”, and “x = y” each fail because at least one admissible case contradicts them. “x ≥ y” also fails because x can be −27, which is less than y.

Common Pitfalls:
Forgetting that a squared equation has two roots; or treating √729 as ±27 (principal square root is +27 only). Mixing these causes incorrect certainty.

Final Answer:
Relationship cannot be determined