Concepts
1 BasicFunctions
Copyright © Cengage Learning. All rights reserved.
Unit 1A
Review of Operations with
Whole Numbers
Copyright © Cengage Learning. All rights reserved.
1.2
Order of Operations
Copyright © Cengage Learning. All rights reserved.
Order of Operations
The expression 53 means to use 5 as a factor 3 times.
We say that 53 is the third power of 5, where 5 is called the
base and 3 is called the exponent.
Here, 53 means 5  5  5 = 125.
The expression 24 means that 2 is used as a factor 4 times;
that is, 24 = 2  2  2  2 = 16.
Here, 24 is the fourth power of 2.
4
Order of Operations
Just as we use periods, commas, and other punctuation
marks to help make sentences more readable, we use
grouping symbols in mathematics, such as parentheses
“( )” and brackets “[ ],” to help clarify the meaning of
mathematical expressions.
Parentheses not only give an expression a particular
meaning, they also specify the order to be followed in
evaluating and simplifying expressions.
5
Order of Operations
What is the value of 8 – 3  2? Is it 10? Is it 2? Or is it some
other number?
It is very important that each mathematical expression have
only one value.
For this to happen, we all must not only perform the exact
same operations in a given mathematical expression or
problem but also perform them in exactly the same order.
6
Order of Operations
The following order of operations is followed by all.
Order of Operations
1. Always do the operations within parentheses or other
grouping symbols first.
2. Then evaluate each power, if any. Examples:
4  32 = 4  (3  3) = 4  9 = 36
52  6 = (5  5)  6 = 25  6 = 150
7
Order of Operations
3. Next, perform multiplications and divisions in the order in
which they appear as you read from left to right.
For example,
60  5  4  3  2
=
300  4  3  2
=
75  3  2
=
25  2
=
50
8
Order of Operations
4. Finally, perform additions and subtractions in the order in
which they appear as you read from left to right.
Note:
If two parentheses or a number and a parenthesis occur
next to one another without any sign between them,
multiplication is indicated.
By using the above procedure, we find that
8 – 3  2 = 8 – 6 = 2.
9
Example 1
Evaluate: 2 + 5(7 + 6).
= 2 + 5(13)
Add within parentheses.
= 2 + 65
Multiply.
= 67
Add.
Note:
A number next to parentheses indicates multiplication.
In Example 1, 5(13) means 5  13.
Adjacent parentheses also indicate multiplication:
(5)(13) also means 5  13.
10
Order of Operations
If pairs of parentheses are nested (parentheses within
parentheses, or within brackets), work from the innermost
pair of parentheses to the outermost pair.
That is, remove the innermost parentheses first, remove
the next innermost parentheses second, and so on.
11