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Notes on Order of Operations
What are Order of Operations?
It is an order to how we calculate answers. First we simplify grouping symbols, then exponents, then
multiplying and dividing. Finally adding and subtracting.
Why do we have an order (of operations)?
We have an order so that everyone--all across the world--arrives at the same answer for the same given
problem.
Are there any tricks to remembering the order?
Some people remember PEMDAS (see below) when thinking of the Order of Operations, but that can
be a little deceptive as there is a bit more to it. Keep in mind the following notes.
ORDER OF OPERATIONS
(Please Excuse My Dear Aunt Sally—PEMoDAoS)
1. parentheses (grouping)
2. exponents
3. multiplication/division (left to right)
4. addition/subtraction (left to right)
Important notes to keep in mind.
1. The “Parentheses” step includes all grouping symbols, such as (parentheses ), [brackets], fraction
bars separating multi-step numerators and denominators, and absolute value.
2. Order of operations is followed within grouping symbols.
3  4  5 3  20 23


 23
2
10  9 1
Example: 10  3
(Take note of the big fraction bar grouping symbol.)
3. When simplifying “Exponents”, make sure you note whether or not the sign is being squared, cubed,
or raised to a higher power.
(3)2  9
Examples:
32  9
3)2  9
0  32  0  9  9
4. Multiplication and division happen at the same time, from left to right in the problem.
5. Distributing (by multiplying) a number across parentheses is a form of multiplication.
9  3 10  27 10  2.7 (multiplication first)
Examples:
9  310  310  30 (division first)
8(3 x)  8  3 8  x  24  8x (distributing)
6. Taking a negative of a number is like multiplying by negative one.
(5)  (1)(5)  5
Example:
7. Adding and subtracting happen at the same time, starting at the left and moving to right in the
problem.
3 7  4  10  4  6 (adding first)
Examples:
3 7  4  4  4  0 (subtracting first)
Notes on Absolute Value
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Date________________________ Class________
Absolute Value—the distance on a number line from 0. Always positive.
The symbol look like this: x (where x is a number or expression)
Examples
2 = 2
 =5
the absolute value of -2 is 2
the absolute value of 5 is 5

Note: The absolute value symbol acts like ( ) are built into it. So the steps are:
 Subtract 5-8
 Take the absolute value (of -3). The simplified answer is 3.
 
The absolute value of -6 is 6. Then the negative sign is tacked on to the front, so the answer
is -6.
Note: The negative sign in front is often read “the opposite of”. It is like multiplying
by a -1.
Notes to Recall
RECALL SYMBOLS:
division
multiplication
a
, a / b, a  b, b a
b
a  b, a  b, (a)(b), ab
Counting numbers or Natural numbers—1, 2, 3, 4, …
Whole numbers—0, 1, 2, 3, 4, …
Integers—…-3. -2, -1, 0, 1, 2, 3, 4, …