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, power, scientific 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

Input	Display	Comment
	blinker	clears screen
3 + 8 × 4	3+8*4
	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

Input	Display	Comment
	blinker	clears screen
3 + 8 × 4 – 5	3+8*4-5²
	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 operation.
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

Input	Display	Comment
	blinker	clears screen
3 + 8 × 4 –   25	3+8*4-√25
	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

Input	Display	Comment
	blinker	clears screen
3 + 8 × 4 – 5  +  7-2	3+8*4-5²+(7-2)
	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.

Input	Display	Comment
	blinker	clears screen
3	-3²
	-9
	blinker	clears screen
3	(-3)²	Parentheses protect order of operations.
	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.

Input	Display	Comment
	blinker	clears screen
5  6	-56
	-15625
	blinker	clears screen
5  6	(-5)6	Parentheses protect order of operations.
	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.

Input	Display	Comment
	blinker	clears screen
5  7  ÷ 2  4	5*107 ÷ 2*104	Division
	2.5*1011	Answer
	blinker	clears screen
5  7   ÷  2  4	(5*107) ÷ (2*104)	Division Parentheses protect order of operations.
	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