Edited to show!
And of course the corollary:
Testingcubed
Testing<sub>cubed</sub>Given my new-found power taught to me by @Kron:
4^6 = 46
Maybe it’s unfamiliar to me because the last time I took a math class the internet being widely available was still shiny and new? I definitely was never expected to type out math homework, anyway, so had no issues representing superscript and all my math textbooks were printed.
You can also make arbitrary fractions without super/subscripts, like this:
1234567654321⁄98765432123456789
Reply if you don’t believe me. The fraction even shows up in the editor.
I think TI is the only major company that commonly used ^ as an actual button on their calculators at any point. It appears in 1990 on their first graphing calculator, the TI-81. The TI-95 Procalc doesn’t have a ^ button but it does display it onscreen to indicate exponentiation, and that was released in 1986. That same year, HP used it on their first graphing calculator, the HP-28c.
HP (and Casio) have otherwise usually stuck to using yx or something similar.
I’m sure these are not the earliest examples of such use by any stretch, but the symbols calculator manufacturers choose to use can be influential on students, so I figured it was worth mentioning.
I agree with @Hari_Seldon that mathematical expressions are not supposed to confuse or mislead people, just the opposite in fact. Also, who writes normal formulae including both an ÷ sign (recommended never to be used; when have you actually seen it) and also / ? So you know that at best things are straight up not kosher.
Someone speculated above that the obelus was supposed to create a display fraction, but—what?—they fell asleep before typesetting it? Same with ^ for superscript.
That’s it. The ^ is all about how exponentiation is represented in computer programming languages that are typed line-by-line into a computer (or keypunched onto “IBM” punch cards; that’s how old ^ for exponentiation is).
Since so much of math and of communication in general nowadays is mediated by a keyboard, we who do math mostly using computers think of the keyboard representations first, and the old fashioned pencil / paper / chalkboard / whiteboard / textbook typeset equation formats second (if at all).
A sizeable fraction of the “Is this homework / TwitFace puzzle screwed up?” threads we get are all about exactly this disconnect, where somebody translates from one format to the other and introduces an unnoticed error along the way.
As I often point out to my students, it should be PERMDAS. Radicals are the opposite operation to exponents and thus belong right next to them.
Agree that 3a is surrounded by implied parenthesis. Which is why your much better mnemonic should always be used:
Though with @John_DiFool’s R for radicals inserted in (hopefully) the right spot.
Overall @glowacks underlying point is really the moral of the story for the whole thread.
Since the goal generally ought to be to be absolutely unambiguous, the right way is to make all evaluation ordering explicit. If the goal is to produce a tricky puzzle, then leaving off the guardrails so a few people spin out and crash in the weeds is a deliberate part of the fun.
Aiming for Door #1 while actually delivering Door #2 is the sign of bad workmanship. Sucks when that’s what’s on the test.
This, unfortunately, is standard for kids’ homework. And not all kids make the leap that a fraction is essentially an “unfinished” division problem, and probably exactly because of situations like this, so they then end up with that “I hate fractions!” mentality.
My mnemonic may do a better job of explicitly explaining the whole order of operations, but I don’t know if that makes it a better mnemonic, because it utterly fails at the “being memorable” part. “Please Excuse My Dear Aunt Sally” is stuck in my head since middle school, I’m not sure how well I will remember
POPE E, M.D., AS AEBETLTBR.
It breaks down at the end there 
The AEBE part isn’t necessary. By definition the last thing is a sequence is the tie breaker for all that came before. But we do need @John_DiFool’s R.
I propose:
POPEER MDAS TLBR. Pronounced “POPEER” is like roper, “MDAS” is em-DASS, and “TLBR” is Till-brrr. Of course in the UK they’ll choose to break the syllables differently so it’ll be POPEERM DASTLBR (emphasis on the DAS) which if you squint sounds like the name of a quaint town in the north of England.
Then we sing it to the tune of some kid’s song I haven’t picked out yet. But I can imagine the Sesame Street characters gleefully repeating it until Mom is ready to throw a wine bottle through her $4,000 TV to “Make it stop!! Please, just make it stop!!!11!1”
This link should help.
PEMDAS works fine when the goal is some basic math.
If you are doing quantum mechanics it probably will not suffice.
Again, it is a mnemonic. It is not meant to be the end-all, be-all of mathematical rules. It suffices for simple equations such as in the OP. I’m not even sure it should make it to algebra.
LSLGuy is a pilot and mnemonics are abundant there too:
CIGAR, FLAPS, BLITTS, MIDGET, CAPER and so on.
Do any of them cover everything? No. Do they mean you can fly a plane? No.
They are merely memory devices for some pieces of the whole puzzle.
PEMDAS is fine as far as it goes. It is not perfect. It is certainly limited. But it definitely suffices to show the problem with the equation in the OP.
http://people.math.harvard.edu/~knill/pedagogy/ambiguity/iso1.jpg
http://people.math.harvard.edu/~knill/pedagogy/ambiguity/iso2.jpg
In short, ambiguity is bad. (Also, ÷ is bad if I understand things correctly.)
I did not make that leap-- someone, another kid, pointed it out to me, and it was a paradigm-shifting moment. Why didn’t a teacher ever just tell me that?
The other kid figured it out when he made the connection between fractions and percentages, something else I never made until it was explained to me, because I never “got” the operation to figure a percentage-- that is, until the moment he explained it to me. Then I never forgot.
He was two years younger than I was, but his parents were both math professors.
I was taught BODMAS. (This was back when computers were the sole province of universities and secret government establishments).
By the time we reached the 5th form, it was well understood that it could equally be written as BOMDSA but by then it didn’t really matter.
Without looking at any of the responses, I got 16.
And now looking at what other people said, I don’t understand the confusion.
8 + (4)^2 ÷ (7/2 - 2)
8 + (4)^2 ÷ (3.5 - 2)
8 + (4)^2 ÷ (1.5)
8 + 16 ÷ (1.5)
24 ÷ (1.5)
16
Division comes before addition in order-of-operations.