However, there is a bit of an issue with its integration, I believe. Namely: they give different answers!  I know this because I tried out an example in the first link up there, that of 2^2^2^2^2.  It turns out the answer to this is a bit vague, so differing solutions are not unexpected, but still…

Why? Exponentiation is not associative, unlike addition or multiplication (it might satisfy the Lie algebra equivalent of the Jacobi identity, I dunno), so the order matters. A lot.  Thus, while 2+(3+4) = (2+3)+4, and likewise for multiplication, 2^(3^4) !=  (2^3)^4. This being exponentiation, it really doesn’t. The latter is 4096, the former is 2417851639229258349412352. Juuuuuuuuuuuuuuuuuuuuust a bit different. So without parentheses, things are vague.

Now how about the example in the article? We’ll get to that. Let’s start with 3 2′s. (2^2)^2 = 16, 2^(2^2) = 16. Phew safe. So, 2^2^2 is fine.

2^2^2^2? Given that, Bing outputs the “bottom-up” answer: ((2^2)^2)^2 = 256. W|A outputs the “top-down”: 2^(2^(2^2)) = 65536. Ah ha.

The disparity is growing, but not bad. So adding one more ^2 shouldn’t be too horrible, right?

Wrong.

Exponentiation grows quickly so order reaaaaaallly matters.

2^2^2^2^2? As expected from above, Bing should do 256^2=65536 and it does. W|A? Well, it does 2^65536. This is a big number. How big? Well…

20035299304068464649790723515

602557504478254755697514192650

169737108940595563114530895061

308809333481010382343429072631

818229493821188126688695063647…

…8091458852699826141425030123391

That is a 19729 digit number. Big.

Therein lies the difference, but which is right? Hmm. Most mathematicians think of repeated exponentiation as “top-down”. So, my main question is why isn’t Bing using W|A as it says? Or is there some extra option I need to assemble?

Oh well. Just an odd thing I noticed.