What 2-digit number(s), when squared, produces a number equal to the cube of its sum of digits?

If a number when squared is a cube, then the original number is also a cube.

The only two digit cubes are 27 and 64.

The cube root of 27 = 3, squared = 9 = 2 + 7, so 27 works.

The cube root of 64 = 4, squared = 16 <> 6 + 4,

So 27 is the only answer.