To me this doesn't seem to make much sense.

Let’s look at addition (hyper 1). 9+4=13, 9+3=12 and 9+1=10. 7+5=12, 7+2=9 and 7+3=10. 13-12=1 and 12-9=3, so (x+y)-(x+z)=y-z.

Now let’s look at multiplication (hyper 2). 8 × 6=48 right, and 8×4=32. Also, 8×2=16. I'll make up some more; say 5×7=35, 5×6=30 and 5×1=5. 48-36=16, so 8×6-8×4=8×2. 35-30=5, so 5×7-5×6=5×1. From this it's clear that (x×y)-(x×z)=x×(y-z).

Now let’s look at exponentiation (hyper 3). 4^5=1024, 4^2=16 and 4^3=64. 3^6=726, 3^5=243 and 3^1=3. 1024÷16=64, so 4^5÷4^2=4^3. 726÷243=3 so 3^6÷3^5=3^1. (x^y) ÷ (x^z)=x^(y-z).

One would expect then that with tetration (hyper 3) (x^^z)√(x^^y)=x^^(y-z), and if (x^^z)√(x^^y)≠x^^(y-z) something’s wrong. If 256=2^^4 (and 4=2^^2) then the 4th root of 256 should be 4, which it is, but including to this article 256 doesn't equal 2^^4 at all, which to me makes no sense whatsoever. Wouldn't the logical thing to do be to do what every calculator in the world does already, and do tetration from left to right?

You do everything else from left to right even when it does make a difference like with division and subtraction for example. Doing it from left to right would also make it easier as going from x^^y to x^^(y+1) would only involve putting x^^y to the power of x just like going from x^y to x^(y+1) only involves multiplying x^y by x and going from x×y to x×(y+1) only involves adding x to x×y.

Let’s look at addition (hyper 1). 9+4=13, 9+3=12 and 9+1=10. 7+5=12, 7+2=9 and 7+3=10. 13-12=1 and 12-9=3, so (x+y)-(x+z)=y-z.

Now let’s look at multiplication (hyper 2). 8 × 6=48 right, and 8×4=32. Also, 8×2=16. I'll make up some more; say 5×7=35, 5×6=30 and 5×1=5. 48-36=16, so 8×6-8×4=8×2. 35-30=5, so 5×7-5×6=5×1. From this it's clear that (x×y)-(x×z)=x×(y-z).

Now let’s look at exponentiation (hyper 3). 4^5=1024, 4^2=16 and 4^3=64. 3^6=726, 3^5=243 and 3^1=3. 1024÷16=64, so 4^5÷4^2=4^3. 726÷243=3 so 3^6÷3^5=3^1. (x^y) ÷ (x^z)=x^(y-z).

One would expect then that with tetration (hyper 3) (x^^z)√(x^^y)=x^^(y-z), and if (x^^z)√(x^^y)≠x^^(y-z) something’s wrong. If 256=2^^4 (and 4=2^^2) then the 4th root of 256 should be 4, which it is, but including to this article 256 doesn't equal 2^^4 at all, which to me makes no sense whatsoever. Wouldn't the logical thing to do be to do what every calculator in the world does already, and do tetration from left to right?

You do everything else from left to right even when it does make a difference like with division and subtraction for example. Doing it from left to right would also make it easier as going from x^^y to x^^(y+1) would only involve putting x^^y to the power of x just like going from x^y to x^(y+1) only involves multiplying x^y by x and going from x×y to x×(y+1) only involves adding x to x×y.