It can be written as ²√2 and there's also ³√2 and so on

Yep.

It would be like if we had a stylistic convention that you can drop the second number in an addition and it becomes 1. So "6 +" means "6 + 1."

Now I can trivially make any number, but I'm just 'exploiting' a convenience feature in the notation:

0 = 2 + 2 - 2 - 2

1 = 2 + 2 - 2 - 2 +

2 = 2 + 2 - 2 - 2 + +

Etc