Compare roots: Which is greater: the fourth root of 10 (∜10) or the cube root of 8 (∛8)?

Introduction / Context: Comparing different radicals often reduces to evaluating known perfect powers or estimating via basic properties.

Given Data / Assumptions: Compare ∜10 and ∛8.

Concept / Approach: 8 is a perfect cube, so ∛8 = 2 exactly. Estimate ∜10 ≈ 10^(1/4).

Step-by-Step Solution: ∛8 = 2 (since 2^3 = 8).∜10 ≈ 10^0.25 ≈ 1.78 (since 1.78^4 ≈ 10).Therefore, 2 > 1.78, so ∛8 is greater.

Verification / Alternative check: Quick reference: ∜16 = 2, and since 10 < 16, ∜10 < 2, confirming the comparison.

Why Other Options Are Wrong: ∜10 is smaller; they are not equal; “None of these” is inapplicable.

Common Pitfalls: Confusing indices (3rd vs 4th roots) or assuming higher index always yields a larger value.

Final Answer: ∛8