Within those three groups it doesn’t really matter which ones you do first
It absolutely does matter. You must do exponents after brackets and before multiplication and division, for the precise reason you said that exponents are shorthand for multiplication. In other words, there's 4 groups, not three.
I agree it needs to be more clearly defined
It's clearly defined in any Maths textbook you pick up.
This is the best video out there
Debunked here - she never once refers to an actual Maths textbook!
Mathematica
Maths doesn’t refer to several kinds of math
It refers to all branches of Mathematics.
rules nobody can even agree on!
All the Maths textbooks agree
implicit multiplication
There's no such thing as "implicit multiplication"
Some place it above explicit multiplication and division,
Which is correct, seeing as how we're solving brackets, and brackets always come first.
But if you place it as equal to it’s explicit counterparts, then you’d sweep left to right giving you an answer of 9
Which is wrong.
Since those are both valid interpretations of the order of operations
No, they're not. Treating brackets as, you know, brackets, is the only valid interpretation. "Multiplication" refers literally to multiplication signs, of which there are none in this problem.
But in reality nobody would write an equation like this
Yes they would. a(b+c) is the standard way to write a factorised term.
6⁄2(1+2) ⇒ 6⁄2*3 ⇒ 6⁄6 ⇒ 1
You're more patient than me to go to that trouble! 😂 But yeah, looks good. Just one technicality (and relates to how many people arrive at the wrong answer), the 2x3 should be in brackets. Yes if you had a proper fraction bar it wouldn't matter, but that's what's missing with inline writing, and is compensated for with brackets (and brackets can't be removed unless there's only 1 term inside). In your original comment, it does indeed look like 6/(2x3), but, to illustrate the issue with what you wrote, as soon as I quoted it, it now looks like (6/2)x3 in my comment.
If it was so well defined, then how did two different sets of rules regarding juxtaposition even come to be?
They didn't - neither of them is a rule of Maths.
I Looked up doing factorials and n! = n(n – 1) is used interchangeably with n! = n*(n – 1)
Yeah, there's a problem with some lazy textbook authors, which I talked about here. A term is defined as ab=(axb), and yet many textbooks lazily write it as ab=axb, which is fine if that's the whole expression, but NOT fine if the expression is a/bc (a/(bxc) and a/bxc AREN'T the same thing!), and so we end up with people removing brackets prematurely and getting wrong answers. In other words, in your case, only n!=n(n – 1) and n!=(nx(n – 1)) can be used interchangeably.
I'm building up quite a collection of these, so far from being a once-off issue. In other words, still looking for any ideas on how to fix this
Are you talking about the lack of indentation? I don't know why it does that. I tried using with the same result. I'd fix it if I knew how!
Well, I just had an issue then here with and eventually found backslash worked - I'll give that a go (not sure if I already tried it before).
Just make sure you don't use an e-calculator, nor a Texas Instruments calculator.