Infinite nested radical (repaired): Evaluate S = √(56 + √(56 + √(56 + …))).

Introduction / Context:The database text contained a likely typo “÷ 22”. By the Recovery-First Policy, we restore the standard infinite nested radical form S = √(56 + √(56 + …)). Such expressions are solved by setting S equal to the entire radical and squaring to remove the root.

Given Data / Assumptions:

• S = √(56 + S) with S > 0.
• Convergent positive solution is expected.

Concept / Approach:Let S represent the entire expression. Then square both sides to obtain a quadratic in S. Solve and take the positive root because S is a principal square root value.

Step-by-Step Solution:S = √(56 + S)Square: S^2 = 56 + SBring terms together: S^2 − S − 56 = 0Solve quadratic: Discriminant D = 1 + 224 = 225, √D = 15S = [1 ± 15]/2 ⇒ S = 8 or S = −7Since S is positive, S = 8

Verification / Alternative check:Plug back: √(56 + 8) = √64 = 8, consistent.

Why Other Options Are Wrong:0, 1, 2, 4 do not satisfy S = √(56 + S). Only S = 8 works.

Common Pitfalls:Keeping the negative root −7; nested radicals defined via principal roots yield nonnegative values.

Final Answer:8