The value of [5(81/3 + 271/3)3]1/4 is

Concept: For a positive base a, writing a1/n means the positive nth root of a — the number whose nth power equals a (for example, 23 = 8, so 81/3 = 2). When a whole expression carries an outer exponent, that exponent applies only after everything inside has been fully simplified.

1. Evaluate the two cube roots inside the innermost bracket: 81/3 = 2 (since 23 = 8) and 271/3 = 3 (since 33 = 27).

2. Add these two results: 2 + 3 = 5.

3. Cube the sum, since the bracket carries an outer exponent of 3: 53 = 125.

4. Multiply by the leading factor of 5 outside the bracket: 5 × 125 = 625.

5. Apply the outermost exponent, 1/4, to 625: since 625 = 54, taking the fourth root gives 5.

Cross-check: Combine the steps directly: 5 × (2 + 3)3 = 5 × 53 = 54, and raising 54 to the power 1/4 directly returns 5 — confirming the step-by-step result.