I have two questions concerning the order of operations. The first relates to the history of mathematics, and the second relates to a computational example that I would be interested in solving without the use of a mnemonic to help me recall the method (which is the only method I am currently familiar with).
Question 1: How algebraic expression are solved before the invention of order of operations?
Question 2: And can you please explain it to me in detail and teach me how to solve this expression without mnemonics? Like $5$ × $4+4$ ÷ $2$ × $8+6^2$ × $3$, for example?
First Question
The order of operations is simply convention. It would, in theory, be possible to rework all of mathematics if, for whatever reason, it was decided that the order of operations should be changed. However, having a clearly defined order of operations that everyone abides by is necessary so that mathematics can be followed by everyone, and there are no misunderstandings in the results that we arrive at.
It’s worth also mentioning that the use of the acronyms PEDMAS / BEDMAS / BIDMAS / BODMAS is believed to have started in the late $1700$s to early $1800$s.
However, the beginning of the convention for the order of operations began emerging as consensus in the $1600$s - about $200$ years prior to the emergence of the acronym system.
Second Question
Your second questions asks for how to solve the following:
$5$ x $4+4$÷$2$ × $8+6^2$ × $3$
The first thing to note here is that, in practice, nobody writes out an expression like this, as it is unnecessarily convoluted. Best practice when writing out anything like this is to use brackets to make everything completely clear.
However, if you are looking for a way to solve this, then the first thing to do is solve any terms with a power (index). And so we can replace $6^2$ with $36$ to give us:
$5$ x $4+4$÷$2$ × $8+36$ × $3$
Now we go from left to right and resolve the multiplication and division operations. And so we replace $5$ x $4$ with $20$, replace $4$ ÷ $2$ with $2$, then replace $2$x$8$ with $16$ and replace $36$ x $3$ with $108$.
This gives us:
$20+16+108$
Which gives us our final answer of $144$ by adding up all of the numbers.
Note: It is worth noting that we always solve any expressions contained within brackets first, before doing anything else. However, this question had no brackets and so this wasn’t necessary here.
Note$_2$:once we reduce the problem down to addition and subtraction, then we can work the problem out in any order that we like as long as we pay close attention to where the minus signs are placed.