Puzzle ZHAI 

Workings

Hint
You don't need to use a calculator.

Answer
Cube root of 2.
We have 10^1/10 vs 2^1/3.

Raise each to the power of 30 (since 10 x 3 = 30) to give:

10^{1/10 x 30} vs 2^{1/3 x 30}, which simplifies to:

10³ vs 2¹⁰, which gives:

1000 vs 1024.

Therefore the cube root of 2 is larger. Alternatively we can use a calculator to see that the tenth root of 10 = 1.2589 and the cube root of 2 = 1.2599.