whats this number?

Question

I understand this is a simple problem in mathematics but still i have some doubt in it.

For example,lets start with a small number 2^2^3 .....this is simple , its in fact 2^8

But if we had, 2^3^4^5 , a quite big number ... then whats the number ?

my doubt here is

should i take the power of 4^5 first ..........(1)

OR

take the power of 3^4 first ? ............(2)


(1) gives ==> :2^3^(1024)

(2) gives ==>2^81^5

which way to go ? whats the order ? and why ?

can you justify ?

Answer 1

there is no ambiguity here...

whenever it is written as such...

it is actually

2^(3^(4^5))

because... ((2^3)^4)^5 = (2^3) ^ 4*5 = 2^ (3*4*5) ... which is not an iteration of exponents... in this case... what is the sense of writing it as further exponents when it is simply multiplication of the basic exponent...


this means... do the innermost relationship first... or in this case the highest exponentiation...


§
Edit: I do not believe in np's answer...
If there are no included parentheses... the rightmost exponent is an exponent of the exponent already...

thus you are correct in interpreting.. 2^2^3 as 2^8 ... the caret symbol is not the usual mathematical symbol used anyway... this is just a reaction to the fact that writing in computer has to be linear... no superscripts allowed...
as far as i can interpret it.. 3 is the exponent of the second 2 not of the whole (2^2)...
meaning.. in handwritten form... 3 is actually a superscript of the superscript.. right...


next edit: i starred this question... so that some of my contacts can give some further scrutiny of the subject... please do not make a best answer very immediately... some of them might have better insight on this...

Answer 2

(aË£)ʸ=a⁽ˣʸ⁾
Also
(ab)c=a(bc)
and
ab=ba
so, it doesn‘t make any difference which order you do the exponentiation…. As long as you do it right (2²)³=2⁶, not 2⁸
So, in that context NK's right

But, if you mean something like
. . . . . . . . 5
. . . . . . . 4
. . . . . . 3
. . . . . 2
Sorry... the site does have its limits... so does MS Word
Then your first objective is to figure out what the exponent is for 2,
First is 4^5 = 1024
Second is 3^1024
Whatever you get for the second iteration, that's the power to which you raise 2.... and it IS a BIG number.
And, in that context, DC's right.

The ambiguity exists because it is impossible to tell if the right exponents are exponents of the next left exponent (DC's surmise), or of the base, 2 (NK's surmise... and mine). Depending on ones background, one can interpret the problem differently.

Answer 3

Unless you're given parentheses or it's written out with superscripts, you always go from left to right.

2^2^3 = (2^2)^3 = 2^6
2^3^4^5 = (((2^3)^4)^5) = (2^12)^5 = 2^60