Order of Operations
When an expression contains more than one operation, we must decide
which operation to carry out first. To do so, we use the following
procedure.
Procedure
To Use the Order of Operations to Simplify an Expression
Step 1 Perform operations inside grouping symbols (such as
parentheses), starting with the innermost set of grouping
symbols.
Step 2 Simplify exponents, square roots, and absolute values.
Step 3 Multiply or divide, working in order from left to right.
Step 4 Add or subtract, working in order from left to right.
The grouping symbols referred to in Step 1 include the following:
Symol 
Name 
Example 
( ) 
parentheses 
6 Ã· (2 + 1) = 6 Ã· 3 = 2 
[ ] 
brackets 
12  [8  (4  1)] = 12  [8  3] = 12  5 = 7 

fraction bar 

  
absolute value 
3 Â· 1  5  = 3 Â· 4 = 3 Â·
4 = 12 

radical symbol 

Example 1
Find 5  6^{2} Ã· 2  (9  5) Â· 3
Solution 
5  6^{2} Ã· 2  (9  5) Â· 3 
Step 1 Perform operations inside
grouping symbols.

= 5  6^{2} Ã· 2  4 Â· 3 
Step 2 Simplify exponents, square roots,
and absolute values.

= 5  36 Ã· 2  4 Â· 3 
Step 3 Multiply or divide, working
in order from left to right. 
= 5  18  12 
Step 4 Add or subtract, working in
order from left to right. 
= 25 
So, the result is 25.
Example 2
Find:
Solution 

Step 1 Perform operations inside
grouping symbols.


Step 2 Simplify exponents, square
roots,and absolute values. 
= 3 + 16 Ã· 2 
Step 3 Multiply or divide, working in
order from left to right. 
= 3 + 8 
Step 4 Add or subtract, working in
order from left to right.

= 11 
Thus, the result is 11.
Example 3
Find 20 Ã· 2 [1  (8  12)]
Solution 
20 Ã· 2 [1  (8  12)] 
Step 1 Perform operations inside grouping
symbols, starting with the
innermost grouping symbol.


First, simplify (8  12).
Now simplify [1  (4)]. 
= 20 Ã· 2[1
(4)] = 20 Ã· 2[5]
= 20 Ã· 2 Â· 5 
Step 2 Simplify exponents, square roots,
and absolute values.
There are none to simplify.


Step 3 Multiply or divide, working in order
from left to right.


First, simplify 20 Ã· 2.
Now simplify 10 Â· 5. 
= 10 Â· 5
= 50 
Therefore, the result is 50.
