Here's one more way to see this
Let us suppose that the result of taking the square root is some integer > 12
Then....when we square both sides we have that
144 - (x)^(1/3) = (minimum of 169)
Subtract 144 from both sides
-(x)^(1/3) = (minimum of 25)
Call the right side some positive integer, P
- (x)^(1/3) = P
(x)^(1/3) = -P
(x)^(1/3) = (-1)P cube both sides
x = (-1)^3 P^3
x = - P^3
So.....x is negative.....but we require that x is positive....so.....the result of evaluating the original expression cannot be an integer > 12
Now suppose that is some integer between 0 - 12 inclusive
So.....when we square both sides......the max = 144 and the min = 0
If the max is 144 then
144 - (x)^(1/3) = 144
-(x)^(1/3) = 0
x^(1/3) = 0
And x = 0 so x is non-negative
If the min is 0
144 - (x)^(1/3) = 0
-(x)^(1/3) = - 144
x^(1/3) = 144
x = 144^3 and x is also non-negative
So x will be positive whenever the original quantity under the root is a perfect square and the right side is an integer from 0 to12 inclusive