Solve the radical equation: cube_root(5568 / 87) + sqrt(72 * 2) = sqrt(?). Find the value under the square root (the radicand).

Introduction / Context:
Here different radicals combine in one statement. The expression asks for the radicand (?) such that the sum of a cube root and a square root equals a single square root. Recognizing perfect powers inside the radicals allows exact integer simplification without calculators.

Given Data / Assumptions:

• cube_root(5568 / 87) + sqrt(72 * 2) = sqrt(?).
• cube_root denotes the principal real cube root; sqrt denotes the principal square root.
• All quantities are positive.

Concept / Approach:
Simplify each radical by factoring to known perfect powers. If the left side simplifies to an integer M, then we need sqrt(?) = M, so ? = M^2. This approach avoids squaring both sides at the start and keeps arithmetic clean.

Step-by-Step Solution:
Compute 5568 / 87. Note that 87 * 64 = 5568, so 5568/87 = 64.cube_root(64) = 4.Compute sqrt(72 * 2) = sqrt(144) = 12.Left side = 4 + 12 = 16.If sqrt(?) = 16, then ? = 16^2 = 256.

Verification / Alternative check:
Check: sqrt(256) = 16, which equals cube_root(64) + sqrt(144) = 4 + 12. Equality holds exactly.

Why Other Options Are Wrong:

• 4 / 16: These are values of the simplified radicals themselves, not the required radicand.
• √2 / 128: Do not satisfy sqrt(?) = 16.

Common Pitfalls:
Misreading cube root as square root; forgetting that the question asks for the value under the square root; arithmetic slips when dividing 5568 by 87.

Final Answer:
256