What is PE(MD)(AS)?
PE(MD)(AS) is an achronym that is used to help students remember the order of operations in an
arithmetic expression.
What does PE(MD)(AS) stand for?
• P = Parenthesis
• E = Exponents
• M = Multiplications
• D = Divisions
• A = Additions
• S = Subtraction
Why does PE(MD)(AS) have parenthesis on its name?
The letters in PE(MD)(AS) tell us about the order in which the operations have to be performed.
Parenthesis first (but starting with the inner most), then Exponents (again the inner most). The
parenthesis around MD is to remind the student that after the Exponents, Multiplications or Divisions
are done, whichever comes first in the expression, when the expression is scanned from LEFT to
RIGHT. Again, once that all the Multiplications and Divisions have been done, the turn is for Addition
or Subtractions, whichever comes first in the expression, when it is scanned from LEFT to RIGHT.
How does PE(MD)(AS) work?
Given an arithmetic expression, PE(MD)(AS) it is used many times on the expression until it is
reduced to a simple expression where there is no doubt of what to do in order to get the numeric
value of the expression. Let us work out an example:
Find the value of the following arithmetic expression:
3 − 4 + 5(3 − (16 ÷ 4 × 2 − 6)2 + 2)
We do Parenthesis first, but we start we the inner most parenthesis, in this case is the expression
16 ÷ 4 × 2 − 6 . To work out this expression we apply again PE(MD)(AS) to it, observe that it won’t have
Parenthesis since we took the inner most parenthesis, so we have to look for Exponents, which in this
case there are none. The next step is to look for Multiplications or Division from LEFT to RIGHT of the
expression, in this case we have to do 16 ÷ 4 , which is 4, so our expression 16 ÷ 4 × 2 − 6 got
simplified to 4 × 2 − 6 . Since this is still an expression (not a number) we have to keep applying
PE(MD)(AS). No P, no E, but we still have (MD) so we proceed from LEFT to RIGHT, now is the turn
of 4 × 2 which is 8, so our expression got simplified to 8 − 6 , which is 2. Now the original expression
got simplified to
3 − 4 + 5(3 − 2 2 + 2)
Since we still have Parenthesis, we have to work the expression: 3 − 2 2 + 2 . No P, but we have
Exponents so we have to do them first, in this case we have to do 2 2 which is 4, so our expression got
simplified to 3 − 4 + 2 . Since this is still an expression we have to repeat the process. No P, no E, no
(MD), but we have Additions or Subtractions and we have to do whichever comes first from LEFT to
RIGHT, in this case we have to do 3 − 4 , which is equal to -1. So our expression got simplified to
−1+ 2 , which is a simple expression and is equal to 1. The original expression now is 3 − 4 + 5 × 1 ,
since this is still an expression we keep applying PE(MD)(AS). No P, no E, but we have (MD).
Remember we must do whichever comes first from LEFT to RIGHT in the expression, in this case is
the turn for 5 × 1 which is 5. Our expression now is: 3 − 4 + 5 . Again this is an expression not a
number, so we have to repeat PE(MD)(AS) again. No P, no E, no (MD), but we have (AS), we have to
do whichever comes first from LEFT to RIGHT, so we must do: 3 − 4 which is equal to -1. Finally our
original expression got simplified to
−1+ 5 which can be easily computed and get 4, which is the value of the original expression.