Compute the exact value of √248 + √(52 + √144). Express the final result numerically (rounded to two decimals if needed) and clearly show your steps.

Introduction / Context:
This arithmetic problem involves nested square roots and evaluates to a real number. It tests the ability to simplify square roots step by step and handle nesting correctly.

Given Data / Assumptions:

• Expression: √248 + √(52 + √144).
• Use exact simplification where possible (e.g., √144 = 12).
• Round to two decimals if the result is irrational.

Concept / Approach:
Simplify inner radicals first, then compute the outer roots. Recognize perfect squares to keep the computation clean. Combine results at the end.

Step-by-Step Solution:
Compute √144 = 12. Then 52 + √144 = 52 + 12 = 64. So √(52 + √144) = √64 = 8. Next evaluate √248. Note 248 = 4 * 62, so √248 = √4 * √62 = 2√62 ≈ 2 * 7.8740 = 15.748. Finally, √248 + √(52 + √144) ≈ 15.748 + 8 = 23.748. Rounded to two decimals: 23.75.

Verification / Alternative check:
A calculator check: √62 ≈ 7.8740; doubling gives 15.748. Add 8 to get 23.748, matching our rounded 23.75.

Why Other Options Are Wrong:
22.00 and 20.00 are too low; they ignore the exact contribution from √248. 24.00 is close but slightly high; the exact computation yields approximately 23.75, not 24.

Common Pitfalls:
Misreading √(52 + √144) as √52 + √144 or approximating √248 as √256 (16) leads to incorrect answers. Always simplify the nested radical correctly before summing.

Final Answer:
23.75