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PEMDAS
The Order of Operations
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1
Introduction
Determine the value of each expression:
1. 18 + 12÷6
2. 18 − 12 · 6 + 6
3. 18 ÷12 ÷6
4.
5−5·9
5·9+8·4
5.
(5)2 · [6 − (4) · 5]
25 · 9 + 8 · 4
2
6. 23
By now you should have no trouble taking two numbers and adding/subtracting/multiplying/dividing them.
But when there are more than two numbers the situation gets a bit more complicated.
When an expression or formula has multiple operations it’s possible for you to get different answers depending
on the order you made the calculations.
2
Find a second way to calculate 23 from problem 6. Next, type 3^2^2 into your calculator. What answer
does it give? What can you type into the calculator to get the other answer? Obviously it’s unacceptable two
have two different answers for the same problem mathematicians came up with rules governing the order in
which the calculations should be performed. Carry out the calculations using the rules and you’ll get the correct
answer. If you carry out the answers in a different order you may or may not get the correct answer. Knowing the
order of operations is very important so you will need to memorize them. When you solve problems, you need
to follow the rules. And when you write mathematics your work must be correct under the order of operations.
2
Finally, as we see with 23 you need to know the rules in order to use a calculator properly. Knowing the rules is
very important!
2 PEMDAS
2
2
PEMDAS
The rules governing the order of operations are often called PEMDAS, for short.
PEMDAS
PEMDAS is an acronym for:
Parentheses
Exponentiation
Multiplication
Division
Addition
Subtraction
1. Parentheses and other grouping symbols such as (, {, [, or | should be calculated first from left to
right.
2. Exponents should be calculated second, from left to right.
3. Multiplication and Division are considered of equal importance and they should be calculated out
from left to right.
4. Addition and Subtraction are considered of equal importance and they should be carried out from
left to right.
While PEMDAS is an acronym for “Parentheses, Exponentiation, Multiplication, Division, Addition, and
Subtraction” people remember PEMDAS with the sentence, “Please Excuse My Dear Aunt Sally”.
2
PEMDAS doesn’t tell us how to evaluate 23 . This is is just one example where there is no universal
agreement on the order operations.
2
The problem 23 is usually interpreted by calculating the exponents from top to bottom (or right to left) to get
29 = 512 but again, this is not universally agreed upon. So be aware that another calculator might give a different
answer to the same problem. If you encounter such an example then you will need to clarify what is meant. It
also means that you should write the problem in a way that makes it clear how interpreted the problem.
If you write 3 − 8 · 4 then it’s assumed you mean 3 − 32 = −29 because that’s what it should evaluate to
using the rules of PEMDAS. Write (3 − 8) · 4 if you want 3 − 8 to be calculated first, because parentheses have
to be calculated first. The answer is then −5 · 4 = 20.
You need to write mathematics which is clear and unambiguous. Use grouping symbols to force the reader to
calculate the way you want them.
3 Exercises
3
Exercises
Use PEMDAS to determine the value of the following expressions.
1. 5÷5÷9÷6
2.
(3.2 − 2.5)÷8
2.8÷10
3.
(36)2 (3.2 + 2.8)
(2.9 − 2.6)50
4.
|3.6 − 2.7 − 1.6|
3.2÷2.5÷2.8
5.
6 · 8 + 11 · 4
(3.2)2
3
4 Solutions
4
4.1
4
Solutions
Introduction
1. 18 + 12÷6 = 18 + 2 = 20
2. 18 − 12 × 6 + 6 = 18 − 72 + 6 = −54 + 6 = −48
1 1
· 6 = 69 · 16 =
3. 18 ÷12 ÷6 = 18 · 12
1
4
4.
5 − 45
40
5−5·9
=
=−
5 · 9 + 8 · 4 45 + 32
77
5.
(5)2 · [6 − (4) · 5] (25) · [6 − (20)] (25) · [−14]
350
=
=
=−
25 · 9 + 8 · 4
25 + 288
313
313
2
6. 23 = 512
4.2
Exercises
1
1
1
1
1. 5÷5÷9÷6 = 5
=
5
9
6
54
2.
(3.2 − 2.5)÷8 (0.70)÷8 (7) · 10
5
=
=
=
2.8÷10
2.8÷10
28 · 8
16
3.
(36)2 (3.2 + 2.8) 1296 · (60) 2592
=
=
(2.9 − 2.6)50
3(50)
5
4.
|3.6 − 2.7 − 1.6|
| − 0.70|
0.70 · 25 · 28
49
= 32 100 =
=
3.2÷2.5÷2.8
32 · (100)
320
25 · 28
5.
6 · 8 + 11 · 4 48 + 44 100 ∗ (92) 575
=
=
=
32 2
(3.2)2
1024
64
10