Why is exponentiation applied right to left?

I am reading an Intro to Python textbook and came across this line:

Operators on the same row have equal precedence and are applied left to right, except for exponentiation, which is applied right to left.

I understand most of this, but I do not understand why they say exponentiation is applied right to left. They do not provide any examples either. Also, am I allowed to ask general questions like this, or are only problem solving questions preferred?

Solution 1:

The ** operator follows normal mathematical conventions; it is right-associative:

In the usual computer science jargon, exponentiation in mathematics is right-associative, which means that x^yz should be read as x^(yz), not (x^y)^z. In expositions of the BODMAS rules that are careful enough to address this question, the rule is to evaluate the top exponent first.

and from Wikipedia on the Order of Operations:

If exponentiation is indicated by stacked symbols, the usual rule is to work from the top down, because exponention is right-associative in mathematics.

So 2 ** 3 ** 4 is calculated as 2 ** (3 ** 4) (== 2417851639229258349412352) not (2 ** 3) ** 4 (== 4096).

This is pretty universal across programming languages; it is called right-associativity, although there are exceptions, with Excel and MATLAB being the most notable.

Solution 2:

from http://docs.python.org/reference/expressions.html

Operators in the same box group left to right (except for comparisons, including tests, which all have the same precedence and chain from left to right — see section Comparisons — and exponentiation, which groups from right to left).

>>> 2 ** 2 ** 2
16
>>> 2 ** 2 ** 2 ** 2
65536
>>> (2 ** 2 ** 2) ** 2
256

For the middle case 2 ** 2 ** 2 ** 2, this are the intermediate steps -

1. broken down to 2 ** (2 ** (2 ** 2))
2. 2 ** (2 ** (4)) # progressing right to left
3. 2 ** (16) # this is 2 to the power 16 which finally evals to 65536 Hope that helps!