Evaluate the nested radical expression √(4600 + √(540 + √(1280 + √(250 + √36)))).

Introduction / Context:
Nested square root expressions are popular in aptitude exams because they look intimidating but often simplify to neat integers if evaluated carefully from the inside out. This problem tests comfort with square roots and recognition of perfect squares hidden inside larger numbers.

Given Data / Assumptions:
We must evaluate:

• S = √(4600 + √(540 + √(1280 + √(250 + √36))).
• All roots are principal square roots.
• The structure suggests that each nested expression becomes a perfect square at each step.

Concept / Approach:
The strategy is to work from the innermost radical outward. We repeatedly simplify each square root, checking whether the expression inside is a perfect square. If it is, the square root simplifies to an integer, which then feeds into the next level. These carefully designed chains commonly collapse to an integer at each step.

Step-by-Step Solution:
Start with the innermost term: √36 = 6. Next layer: √(250 + √36) = √(250 + 6) = √256. Since 256 = 16^2, √256 = 16. Move one level out: √(1280 + 16) = √1296. 1296 is 36^2, so this root equals 36. Next level: √(540 + 36) = √576. 576 is 24^2, so the value is 24. Finally evaluate the outermost root: √(4600 + 24) = √4624. 4624 is 68^2, so √4624 = 68. Therefore the overall value of the nested radical is 68.

Verification / Alternative check:
We can verify by squaring the final answer: 68^2 = (70 - 2)^2 = 70^2 - 2*70*2 + 2^2 = 4900 - 280 + 4 = 4624. This matches the outermost inner value 4600 + 24. Since each inner radical was also checked as a perfect square, the entire computation is consistent.

Why Other Options Are Wrong:
69, 70 and 72 correspond to nearby squares 4761, 4900 and 5184, all of which are larger than 4624. 64 gives 4096 when squared, which is too small. These distractors would arise if a step such as 540 + 36 were miscomputed, or if a root like √576 were misread as 25 instead of 24.

Common Pitfalls:
A common mistake is to rush and treat 250 + 36 as 286 or misread 1296 as 34^2 rather than 36^2. Another issue is not recognising perfect squares like 256, 576 and 4624 quickly. Practising squares of integers up to 50 makes such questions much faster and more reliable.

Final Answer:
The value of the nested radical is 68.