Evaluate √289 − (√625 ÷ √25) using operator precedence (division before subtraction).

Introduction / Context:
This short radical arithmetic problem checks whether you apply operator precedence correctly. You must evaluate the division within radicals before performing the subtraction. The values here are perfect squares, so all square roots simplify to integers first.

Given Data / Assumptions:

• Expression: √289 − √625 ÷ √25.
• Operator precedence: multiplication/division before addition/subtraction.
• All square roots denote principal (nonnegative) values.

Concept / Approach:
Compute each square root exactly, then perform the division, and finally the subtraction. Since all numbers are perfect squares, the operations reduce to simple integer arithmetic, making mistakes mostly about order rather than complexity.

Step-by-Step Solution:
Simplify roots: √289 = 17; √625 = 25; √25 = 5.Apply precedence: first 25 ÷ 5 = 5.Now subtract: 17 − 5 = 12.

Verification / Alternative check:
Rewrite as 17 − (25/5) = 17 − 5 = 12 — confirms the same result.

Why Other Options Are Wrong:

• 17 or 15: Arise from subtracting before dividing, or miscomputing a square root.
• -8/5: Comes from mishandled order and sign together.
• 10: A common incorrect mental slip (17 − 7) not present here.

Common Pitfalls:
Doing subtraction before division; treating √625 ÷ √25 as √(625/25) without confirming both are positive (it is valid here, still giving 5, but many misapply in other contexts).

Final Answer:
12