Order of Operations

Order of Operations = PEMDAS


Order of Operations
 go in order of priority by the acronym PEMDAS:
1. Parentheses
2. Exponents (Powers and Roots)
3. Multiplication and Division
4. Addition and Subtraction

When operations have the same priority, the operations move from left to right.

For reasons which will become obvious in the Answer Processes below, a question involving order of operations is usually better suited to the calculator.

However, even with the calculator, you may need to use parentheses to protect the order of operations in calculations involving squaring, powerscientific notation, and other potentially confusing mathematical situations.


Question

What is the value of 3 + 8 × 4?

Answer
35

Answer Process
The multiplication operation has the highest priority in this mathematical expression.
8 × 4 = 32
Rewrite the expression.
3 + 8 × 4 = 3 + 32
3 + 32
The addition operation is all that remains.
3 + 32 = 35

Calculator Click What You See Comment
clear blinker clears screen
3 + 8 × 4 3+8*4
enter 35 Answer
Order of Operations


Question

What is the value of 3 + 8 × 4 – 5²?

Answer
10

Answer Process
The exponent operation has the highest priority in this mathematical expression.
5² = 25
Rewrite the expression.
3 + 8 × 4 – 5² = 3 + 8 × 4 – 25
3 + 8 × 4 – 25
The multiplication operation has the highest priority in the rewritten expression.
8 × 4 = 32
Rewrite the expression.
3 + 8 × 4 – 25 = 3 + 32 – 25
3 + 32 – 25
Because the addition operation has the same priority as the subtraction operation in the rewritten expression, perform the operation on the left.
3 + 32 = 35
Rewrite the expression.
3 + 32 – 25 = 35 – 25
35 – 25

The subtraction operation is all that remains.
35 – 25 = 10

Calculator Click What You See Comment
clear blinker clears screen
3 + 8 × 4 – 5 square 3+8*4-5²
enter 10 Answer
Order of Operations


Question

What is the value of 3 + 8 × 4 – 25 ?

Answer
30

Answer Process
The root operation has the highest priority in this mathematical expression.
25 = 5
Rewrite the expression.
3 + 8 × 4 – 25 = 3 + 8 × 4 – 5
3 + 8 × 4 – 5
The multiplication operation has the highest priority in the rewritten expression.
8 × 4 = 32
Rewrite the expression.
3 + 8 × 4 – 5 = 3 + 32 – 5
3 + 32 – 5
Because the addition operation has the same priority as the subtraction operation in the rewritten expression,
perform the operation on the left.
3 + 32 = 35
Rewrite the expression.
3 + 32 – 5 = 35 – 5
35 – 5

The subtraction operation is all that remains.
35 – 5 = 30

Calculator Click What You See Comment
clear blinker clears screen
3 + 8 × 4 – 2nd square root 25 3+8*4-√25
enter 30 Answer
Order of Operations


Question

What is the value of 3 + 8 × 4 – 5² + (7 – 2)?

Answer
15

Answer Process
The operation in parentheses has the highest priority in this mathematical expression.
(7 – 2) = 5
Rewrite the expression.
3 + 8 × 4 – 5² + (7 -2 ) = 3 + 8 × 4 – 5² + 5
3 + 8 × 4 – 5² + 5
The exponent operation has the highest priority in the rewritten expression.
5² = 25
Rewrite the expression.
3 + 8 × 4 – 5² + 5 = 3 + 8 × 4 – 25 + 5
3 + 8 × 4 – 25 + 5
The multiplication operation has the highest priority in the rewritten expression.
8 × 4 = 32
Rewrite the expression.
3 + 8 × 4 – 25 = 5 = 3 + 32 – 25 + 5
3 + 32 – 25 + 5
Because the addition, subtraction, and addition operations have the same priority in the rewritten expression, perform the operation on the left.
3 + 32 = 35
Rewrite the expression.
3 + 32 – 25 + 5 = 35 – 25 + 5
35 – 25 + 5
Because the subtraction operation has the same priority as the addition operation in the rewritten expression, perform the operation on the left.
35 – 25 = 10
Rewrite the expression.
35 – 25 + 5 = 10 + 5
10 + 5
The addition operation is all that remains.
10 + 5 = 15

Calculator Click What You See Comment
clear blinker clears screen
3 + 8 × 4 – 5 square + ( 7-2 ) 3+8*4-5²+(7-2)
enter 15 Answer
Order of Operations


Question
According to the calculator, what is the value of -3²?
According to the calculator, what is the value of (-3)²
?
Observe how parentheses change the results of these calculations.

Answer
-3² = -9
(-3)² = 9

Answer Process
The second calculation illustrates the way parentheses protect the order of operations when squaring a negative number.

Calculator Click What You See Comment
clear blinker clears screen
(-) 3 square -3²
enter -9
clear blinker clears screen
( (-) 3 ) square  (-3)²
Parentheses protect order of operations.
enter  9
Order of Operations
Squaring


Question
According to the calculator, what is the value of -56?
According to the calculator, what is the value of (-5)6
?
Observe how parentheses change the results of these calculations.

Answer
-56 = -15625
(-5)6 = 15625

Answer Process
The second calculation illustrates the way parentheses protect the order of operations when working with the power of a negative number.

Calculator Click What You See Comment
clear blinker clears screen
(-) 5 power 6 -56
enter -15625
clear blinker clears screen
( (-) 5 ) power 6 (-5)6
Parentheses protect order of operations.
enter 15625
Order of Operations
Power


Question
According to the calculator, what is the value of 5 × 107 ÷ 2 × 104?
According to the calculator, what is the value of (5 × 107) ÷ (2 × 104)?
Observe how parentheses change the results of these calculations.

Answer
5 × 107 ÷ 2 × 104 = 2.5 × 1011
(5 × 107) ÷ (2 × 104) = 2500

Answer Process
The second calculation illustrates the way parentheses protect the order of operations when scientific notation involves division.

Calculator Click What You See Comment
clear blinker clears screen
scientific notation 7 right ÷
scientific notation 4 right
5*10÷ 2*104 Division
enter 2.5*1011  Answer
clear blinker clears screen
( 5 scientific notation 7 right ) ÷
( 2 scientific notation 4 right )
(5*107) ÷ (2*104) Division

Parentheses protect order of operations.
enter 2500 Answer
Order of Operations
Scientific Notation


Practice – Questions
1.  What is the value of 5 + 12 × 6?

2.  What is the value of 5 + 12 ÷ 6?

3.  What is the value of 5 – 9 ÷ 3 + 81 ?

4.  What is the value of 5 – 9 ÷ 3 + 7² – (6 + 4)?

5.  What is the value 0f -8²?

6.  What is the value (-8)²?

7.  What is the value of -48?

8.  What is the value of (-4)8?

9.  What is the value of 7 × 107 ÷ 4 × 104?

10.  What is the value of (7 × 107) ÷ (4 × 104)?


Practice – Answers
1.  77

2.  7

3.  11

4.  41

5.  -64

6.  64

7.  -65536

8.  65536

9.  1.75 × 1011

10.  1750

About Scott Solomon

HowtoPasstheGED.com gears up adult learners for the recently redesigned, newly computerized GED. ScottSolomon.com delves into my literary fiction.
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