Quote:

so 4^2 = 16 [yes]
4ft^2 = 4 [no]
2 x 2 = 16 [no] because 4^2 = 16 [yes]
2ft x 2ft = 4ft^2 [yes] because 4ft^2 = 4 [no]

As I've said before, the ^2 applies only to the "ft", not to the 4. Your reasoning above doesn't work when read that way. This is the common convention. It's not a natural truth, it's just an arbitrary agreement on how we write math so people can understand each other.

If you want to square the 4 but not the ft, then put the squared symbol on the 4 and not on the ft.

4^2 ft