Evaluate the expression with roots and exponents: (5568 ÷ 87)^(1/3) + (72 × 2)^(1/2) = (?)^(1/2). Find the value of ?.

Introduction / Context:This item tests correct interpretation of fractional exponents and order of operations. You must carefully compute a cube root and a square root, add the results, and then express the sum as the square root of an unknown number (?^(1/2)).

Given Data / Assumptions:

• (5568 ÷ 87)^(1/3) + (72 × 2)^(1/2) = (?)^(1/2).
• Principal (non-negative) roots are intended.
• Exact arithmetic is expected (no rounding needed).

Concept / Approach:Evaluate the division 5568 ÷ 87 exactly, then take its cube root. Next, compute 72 × 2 and take the square root. Add the two results to obtain a value that must equal √?. Finally, square this sum to recover ?.

Step-by-Step Solution:

Verification / Alternative check:Plug back: √256 = 16; left side is 4 + 12 = 16. Both sides match exactly.

Why Other Options Are Wrong:4 and 16 are intermediate root values, not ?. √2 is unrelated. 144 is the product 72 × 2 before taking the square root, not the required ?.

Common Pitfalls:Taking square root before multiplication, or misreading the cube root as a square root. Keep radical operations in the correct order.

Final Answer:256