Prerequisite skill: Order of Operation (PEMDAS)
If you are asked to simplify something like "4 + 2×3", the question that naturally arises is "Which
way do I do this? Because there are two options!":
Choice 1:
4 + 2×3 = (4 + 2)×3 = 6×3 = 18
Choice 2:
4 + 2×3 = 4 + (2×3) = 4 + 6 = 10
It seems as though the answer depends on which way you look at the problem. But we can't have
this kind of flexibility in mathematics; math won't work if you can't be sure of the answer, or if the
exact same problem can calculate to two or more different answers. To eliminate this confusion,
we have some rules of precedence called "order of operations", the "operations" being addition,
subtraction, multiplication, division, exponentiation, and grouping.
A common technique for remembering the order of operations is the abbreviation "PEMDAS",
which is turned into the phrase "Please Excuse My Dear Aunt Sally". It stands for "Parentheses,
Exponents, Multiplication and Division, and Addition and Subtraction". This tells you the ranks of
the operations: Parentheses outrank exponents, which outrank multiplication and division (but
multiplication and division are at the same rank), and these two outrank addition and subtraction
(which are together on the bottom rank). When you have a bunch of operations of the same
rank, you just operate from left to right. For instance, 15 ÷ 3 × 4 is not 15 ÷ 12, but is
rather 5 × 4, because, going from left to right, you get to the division first.
But PEMDAS can generate its own confusion, because students tend to apply the hierarchy as
though all the operations in a problem are on the same "level", but often they're not. Many times it
helps to work problems from the inside out, because often some parts of the problem are "deeper
down" than other parts. The best way to explain this is to do some examples:

Simplify 4
+ 32 .
Do the exponent before trying to add in the 4:
4 + 32 = 4 + 9 = 13

Simplify 4
+ (2 + 1)2.
You have to simplify inside the parentheses before you can take the exponent through:
4 + (2 + 1)2 = 4 + (3)2 = 4 + 9 = 13

Simplify 4
+ [–1(–2 – 1)]2.
Don't try to do parentheses from left to right; instead, work from the inside out:
4 + [–1(–2 – 1)]2
= 4 + [–1(–3)]2
= 4 + [3]2
Study Guide: Order of Operations
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=4+9
= 13
There is no particular significance in the use of square brackets (the "[" and "]" above) instead of
parentheses. Brackets and curly-braces (the "{" and "}" characters) are used when there are
nested parentheses, as an aid to keeping track of which parentheses go with which. The different
grouping characters are used for convenience only. Reserved
Here are some more examples:

Simplify 4(
–2
/3 + 4/3 ).
Remember to simplify inside the parentheses first:
–2
Then 4(

/3 + 4/3) = 8/3
Simplify 4
– 3[4 –2(6 – 3)] ÷ 2.
Simplify from the inside out: first the parentheses, then the square brackets. Also, be
careful with taking the minus through the parentheses: remember that the –3 goes on
everything inside the brackets!
4 – 3[4 –2(6 – 3)] ÷ 2
= 4 – 3[4 – 2(3)] ÷ 2
= 4 – 3[4 – 6] ÷ 2
= 4 – 3[–2] ÷ 2
=4+6÷2
=4+3
=7
Remember that, in lieu of grouping symbols telling you otherwise, the division comes before the
addition, which is why this simplified in the end as "4 + 3", and not "10 ÷ 2".

Simplify 16
– 3(8 – 3)2 ÷ 5.
Remember to simplify inside the parentheses before you square, because (8 –
2
2
the same as 8 – 3 .
3)2 is not
16 – 3(8 – 3)2 ÷ 5
= 16 – 3(5)2 ÷ 5
= 16 – 3(25) ÷ 5
= 16 – 75 ÷ 5
= 16 – 15
=1
Study Guide: Order of Operations
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The Order of Operations: Examples

Simplify 16
÷ 2[8 – 3(4 – 2)] + 1.
16 ÷ 2[8 – 3(4 – 2)] + 1
= 16 ÷ 2[8 – 3(2)] + 1
= 16 ÷ 2[8 – 6] + 1
= 16 ÷ 2[2] + 1 (**)
= 16 ÷ 4 + 1
=4+1
=5
The confusing part in the above calculation is how "16 divided by 2[2] + 1" (in the line marked
with the double-star) becomes "16 divided by 4 + 1", instead of "8 times by 2 + 1". That's
because, even though multiplication and division are at the same level (so the left-to-right rule
should apply), parentheses outrank division, so the first 2 goes with the [2], rather than with the
"16 divided by". That is, multiplication that is indicated by placement against parentheses (or
brackets, etc) is "stronger" than "regular" multiplication.

Simplify 14x
+ 5[6 – (2x + 3)].
If you have trouble taking a subtraction through a parentheses, try turning it into
multiplying a negative 1 through the parentheses (note the highlighted red "1" below):
14x + 5[6 – (2x + 3)]
= 14x + 5[6 – 1(2x + 3)]
= 14x + 5[6 – 2x – 3]
= 14x + 5[3 – 2x]
= 14x + 15 – 10x
= 4x + 15
Study Guide: Order of Operations
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
Simplify the following:
Be careful when simplifying fractions. You cannot cancel a term from part of a sum, only
a stand-alone factor in a product. Do not try to cancel the 3 with the 9, or the 5 with the
20; this is not legitimate.

Simplify the following:
Remember to reduce fractions when you're done. Copyright © 2006 All Rights Reserved

Simplify the following:
Be careful with fractions! Do not try to cancel the
to get 3 underneath.
Study Guide: Order of Operations
2 with anything! You have to add in the 1 first,
Page 4 of 5

Simplify the following:
Do not try to cancel the

Simplify –{2x
4 or 2 into the 12; add the 4 and 2 to get 6. Only then can you cancel.
– [3 – (4 – 3x)] + 6x}.
If you have trouble with subtracting parentheses, convert the subtraction to multiplying a
negative 1 through the parentheses. Remember to simplify at each step, combining like
terms where possible:
–{2x – [3 – (4 – 3x)] + 6x}
= –1{2x – 1[3 – 1(4 – 3x)] + 6x}
= –1{2x – 1[3 – 4 + 3x] + 6x}
= –1{2x – 1[– 1 + 3x] + 6x}
= –1{2x + 1 – 3x + 6x}
= –1{2x + 6x – 3x + 1}
= –1{5x + 1}
= –5x – 1
The most important things to remember with Order of Operations is to work from the inside out,
and to do each step completely and separately. If you try to do two steps at once, or try to do stuff
in your head, you will be much more likely to make mistakes. Take the time to get in the habit of
being careful. The skill will stand you in good stead, especially on tests where you can't afford
mistakes.
Study Guide: Order of Operations
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