What is 6 ÷ 2(3+2)?

So, there has been a debate going around social media as of late.  It deals with the order of operations, and the fact that the vast majority of people seem to forget that each step of the order of operations ends with the phrase "from left to right".  That phrase is irrelevant most of the time, but when it's not irreverent, it's crucial.

Damn you PEMDAS!

So what is 9!/8!(3^2)?  Is it 81 or 1?  Or is it something else entirely?  Now I am going to tread lightly here so as to not enrage too many hearts.

Before I answer this question, it would be a good idea to review the algebraic order of operations, or PEMDAS:

1. Perform all operations inside parentheses using the remainder of the order of operations as your guide to the order.  Any parentheses within parentheses are to be performed from the inside out and from left to right.
2. Perform any and all exponent from left to right.
3. Perform all multiplication and division from left to right.
4. Perform all addition and subtractions from left to right.

Now, we should go over what the exclamation mark here represents.  It is the factorial function, which tells you to multiply together all integers from 1 to whichever number is to the left of the exclamation mark.  So that means 9!=9*8*7*6*5*4*3*2*1 and 8!=8*7*6*5*4*3*2*1.  So the fraction gives us $\frac{9!}{8!}=9$. That's the easy part.

Now what about that 3²? Everyone agrees that it equals 9, but is it in the numerator or denominator? We have to look at the parentheses for the clue.  The term 8!(3²) has what is called implicit multiplication.  This means that there is no symbol telling you to multiply two factors together, but it is well established that multiplication is to be performed.  If you follow my lecture series or have gone through math or physical sciences lately, you'll know that when we have something like 3x, that represents the phrase "Take 3 and multiply it by the variable x", even though there's no multiplication symbol.  That's the nature of informally implicit multiplication.

So the expression 9!/8!(3²), namely 8!(3²).  Instead of representing it this way, let's represent it with explicit multiplication, where the multiplication symbol is explicitly stated instead of implied.  So the expression is $9!/8! \times (3^{2})$.  When the implicit multiplication is in the form of something multiplied by a thing inside a parentheses, then treat it as if it has a multiplication symbol representing the multiplication.

Now it is a whole lot easier to see where the "from left to right" part comes into play when we're doing the "Perform all multiplication and division from left to right". The division occurs first because it appears first.  That means $9!/8! \times (3^{2})=9/1 \times 9=9 \times 9 =81$.  So 81 is the correct answer, because from the "left to right" clause of step 3 of the order of operations, $9!/8! \times (3^{2})= (9!/8!) \times (3^{2})=  \frac{9!}{8!} \times (3^{2})$.  Now, if they added another set of parentheses so that we had $9!/[8! \times (3^{2})]$, then it would equal 1, because at least then, the $8! \times (3^{2})$ would fall under step 1 of the order of operations.  But since those outer parentheses are not present, the answer is 81 as I have thoroughly explained.

If you have any questions, please leave them in the comments.  If you disagree with my conclusion and my logic, please explain why you disagree in the comments.  But please, everyone try to be civil.

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