Order of Operations ( or PEMDAS Rule ) 
The key concept behind the order of operations is to perform arithmetic operators in the "right" order or sequence. Let's take a look at how Rob and Patty tried to simplify a given numerical expression by applying the order of operations.
Rob 
= 6 + 4 x 3 −10 ÷ 5
=10 x 3 − 10 ÷ 5
= 30 − 10 ÷ 5
= 20 ÷ 5
= 4 
Patty 
= 6 + 4 x 3 −10 ÷ 5
= 6 + 12 −10 ÷ 5
= 6 + 12 −10 ÷ 5
= 6 + 12 −2
=16 
What is Rob's mistake?
 He carelessly simplified the numerical expressions by applying arithmetic operations from left to right.


Patty got the correct answer because she properly applied the rules on order of operations.
 She performed multiplication and division first before addition and subtraction.


To avoid different answers like what happened to Rob and Patty, mathematicians decided to agree on certain rules and procedures to follow in simplifying or calculating numerical expressions. That day, the Order of Operations or PEMDAS rule was created... 
The order of operations or PEMDAS is simply a rule that prioritizes the sequence of operations starting from the most important to the least important.
Step 1: Do as much as you can to simplify everything inside the parenthesis first
Step 2: Simplify every exponential number in the numerical expression
Step 3: Multiply and divide whichever comes first, from left to right
Step 4: Add and subtract whichever comes first, from left to right
Let's take a look at some examples...
Example 1: Simplify 5 ÷ 5 + 3 − 6 x 2 using the rules for order of operations 
= 5 ÷ 5 + 3 − 6 x 2 
Divide 
=1 + 3 − 6 x 2 
Multiply 
=1 + 3 − 12 
Add 
= 4 − 12 
Subtract 
= −8 
Final answer 

Example 2: Simplify 3 x 7 − 11 + 15 ÷ 3 using the rules for order of operations 
= 3 x 7 − 11 + 15 ÷3 
Multiply 
= 21− 11 + 15 ÷3 
Divide 
= 21− 11 + 5 
Subtract 
= 10 + 5 
Add 
= 15 
Final answer 

The next examples will now involve parentheses. Remember that you have to simplify everything inside the parenthesis first before going forward. The rules for order of operations apply the same way inside the parenthesis.
Example 3: Simplify 25 − (7 − 12 ÷ 6 ) x 4 using the rules for order of operations 
= 25 − (7 − 12 ÷ 6) x 4 
Simplify first everything inside the parenthesis 
= 25 − (7 − 12 ÷ 6) x 4 
Divide 
= 25 − (7 −2) x 4 
Subtract 
= 25 − (5) x 4 
Multiply 
= 25 −20 
Subtract 
= 5 
Final answer 

Example 4: Simplify 5(4 + 3 x 2) − (9 −28 ÷ 7 ) ÷ 5 using PEMDAS rule. 
= 5(4 + 3 x 2) − (9 −28 ÷ 7 ) ÷ 5 
Simplify first the expressions inside the parenthesis 
= 5(4 + 3 x 2) − (9 −28 ÷ 7 ) ÷ 5 
Multiply, Divide 
= 5(4 + 6) − (9 − 4 ) ÷ 5 
Add, Subtract 
= 5(10) − (5) ÷ 5 
Multiply 
= 50 − (5) ÷ 5 
Divide 
= 50 −1 
Subtract 
= 49 
Final answer 

The final examples will involve exponents. Be careful in each step because they are so many things going on. As long as you remain focus in following the rules governing the order of operations, it shouldn't be that difficult! Here we go...
Example 5: Simplify 2^{4}−5(10−4^{2} ÷ 2) + (30−3^{3}) using PEMDAS rule. 
= 2^{4}− 5 (10 − 4^{ 2} ÷ 2 ) + (30− 3^{3 }) 
Simplify first the expressions inside the parenthesis, and the exponential numbers. 
= 2^{4}− 5 (10 − 4^{2} ÷ 2 ) + (30− 3^{3}^{ }) 
Exponents 
= 16 − 5 (10 −16÷ 2 ) + (30−27) 
Divide 
= 16 − 5 (10 − 8) + (30−27) 
Subtract 
= 16 − 5 (2) + (3) 
Multiply 
= 16 − 10 + (3) 
Subtract 
= 6 + (3) 
Add 
= 9 
Final answer 

Example 6: Simplify (32 − 3^{ 3 } ÷ 9 x 10)^{5 }− 4^{2} ÷ 8 + 3^{2} using PEMDAS rule. 
=(32 − 3 ^{ 3 } ÷ 9 x 10 )^{5 }− 4^{2} ÷ 8 + 3^{2} 
Simplify first the expressions inside the parenthesis, and numbers with exponents 
=(32 − 3 ^{ 3}^{ } ÷ 9 x 10 )^{5 }− 4^{2} ÷ 8 + 3^{2} 
Exponents 
=(32 − 27 ÷ 9 x 10 )^{5 }−16÷ 8 + 9 
Divide 
=(32 − 3 x 10 )^{5 }−16÷ 8 + 9 
Multiply 
=(32 −30)^{5 }−16÷ 8 + 9 
Subtract 
=(2)^{5}^{ }−16÷ 8 + 9 
Exponent 
=32 −16÷ 8 + 9 
Divide 
=32 −2+ 9 
Subtract 
= 30+ 9 
Add 
= 39 
Final answer 


