Puzzle ZHAI 

Workings

Hint
You don't need to use a calculator.

Answer
Cube root of 2.

Reasoning
We could use a calculator to see that the tenth root of 10 = 1.2589 and the cube root of 2 = 1.2599.

But, if you've not got your calculator handy …

We have:

   10^1/10 or 2^1/3

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

   (10^1/10)³⁰ or (2^1/3)³⁰

   10^{1/10 x 30} or 2^{1/3 x 30}

   10³ or 2¹⁰

   1000 or 1024

Therefore, the cube root of 2 is larger.