# Order of Operations: 360 or 354?

Last Saturday, I received  Facebook message from a student asking help to simplify $[5(4)^3 + 6(11-4)] - 36 / 9 (2)$.  He got $354$ but his teacher’s answer was $360$.

Although the problem above seems simple, a lot of students get confused by it, and in this case, even the teacher too. The expression above simplifies to

$[5(64) + 6(7)] - 36 / 9*2$
$= [320 + 42] - 4(2)$
$=[320 + 42]- 4(2)$
$=362 - 8$
$= 354$.

The  misconception about the order of operations usually arises from acronyms like PEMDAS (parenthesis, exponent, multiplication, division, addition, and subtraction). Mathematical operations should be performed in that order: simplify the expression within the parenthesis first, and then simplify the expressions with exponent, perform multiplication, and so on.  In multiplication and division, however, even though multiplication comes first, if the two operations are adjacent and without parenthesis, we perform the operation from left to right. In the example above, the INCORRECT way to simplify $36/9*2$ is to multiply $9$ by $2$ first before dividing; that is, $36/18=2$. The CORRECT way is dividing $36$ by $9$ first (which equals $4$), and then multiplying it by $2$, which equals $8$.

Note, however, that $36 / 9*2 = 8$, but $36/(9*2)$ is $2$ since you have to simplify the operation within the parenthesis first.

Hence $16/2*2 + 1 = 17$ and NOT $5$, and $6/2*3 = 9$ and NOT $1$.

The rule is you perform mathematical operations in the following order:

1. terms inside parentheses or brackets from the inner set of symbols to the outer set
2. exponents and roots
3. multiplication and division as they appear from left to right
4. addition and subtraction as they appear from left to right

As shown in the list above, addition and subtraction is also performed whichever comes first.