Some Background: Why have an order of operations at all?
The order of operations exists:
- To be able to write down the same expression in different ways.
- So that everyone will (still) arrive at the same answer.
The same calculation can be written in different ways: 1+2*2, or 2*2+1, but it should always give the same answer. The problem is that the answer will depend on how you interpret it, and that will influence how you solve it. If you just go left-to-right and joining two and two numbers, you will get (1+2)*2 = 3*2 = 6 in the first case and (2*2)+1 = 4+1 = 5 in the second case.
So mathematicians have agreed on a standard way of interpreting the expression (what parts belong together and not), independently of how it is written down. This standard way is called "The order of operations", which says you should solve an equation in this order:
- Parenthesis
- Exponentials
- Multiplication and division
- Addition and subtraction
But why are the operations in the order they are in?
The order of operations is ordered the way it is simply by convention (agreement). An author could have used parenthesis for every term of an expression, to specify precisely how he intended the expression to be calculated. Instead, a default order was agreed upon, so that one may remove parentheses, and still have people interpret the expressions in the same manner. The default order of operations is arbitrary. The order could have been different, and worked quite as well.[1]
History: "I would say that the rules actually fall into two categories: the natural rules (such as precedence of exponential over multiplicative over additive operations, and the meaning of parentheses), and the artificial rules (left-to-right evaluation, equal precedence for multiplication and division, and so on). The former were present from the beginning of the notation, and probably existed already, though in a somewhat different form, in the geometric and verbal modes of expression that preceded algebraic symbolism. The latter, not having any absolute reason for their acceptance, have had to be gradually agreed upon through usage, and continue to evolve." - http://mathforum.org/library/drmath/view/52582.html
The natural rules arose naturally, and people used it in the same way. Probably because it is intuitive to solve the most grouped terms of the expression first, so you get a bunch of more primitive terms which can then be added together later.
Exponentials are just grouped multiplication: 5^3 = 5*5*5
Multiplication is just grouped addition: 5*3 = 5+5+5
So, exponentials are grouped multiplication which is grouped addition. When you solve exponential terms out all the way, you are just left with a bunch of numbers which are added together.
[1] The important thing is that the author and the reader of an expression has the same interpretation. If you calculated addition before multiplication, then an expression like 5*5+1 would suddenly mean something entirely different, if the author had presupposed multiplication before addition. If the author knew that the reader would interpret and calculate addition before multiplication (if there were no parentheses to explicitly state the order), he would have to write it as (5*5)+1, to force the reader to multiply before adding. And vice versa.