1.2 Order of Operations

An expression is a combination of numbers and one or more arithmetic operation symbols. Arithmetic operations are addition, subtraction, multiplication, and division. For example, 4+20-7 and (6 \bullet 2)\div20 are expressions. When evaluating expressions, it is important that you calculate the operations in the correct order. 

Consider the expression 4+3 \bullet 7. Two students simplify the expression and get two different answers.
Table 1.2.1
Harmony Masoomeh
4 + 3 \bullet 7
7 \bullet 7
49		 4 + 3 \bullet 7
4 + 21
25

It seems that each student interpreted the problem differently, resulting in two different answers. Harmony performed the operation of addition first, then multiplication, whereas Masoomeh performed multiplication first, then addition. When performing arithmetic operations there can only be one correct answer. We need a set of rules in order to avoid this kind of confusion. Mathematicians have devised a standard order of operations for calculations involving more than one arithmetic operation.

Table 1.2.2
Order of Operations
When simplifying mathematical expressions, perform the operations in the following order:

Step 1: Simplify all expressions inside the brackets.

Step 2: Simplify all expressions with exponents and square roots.

Step 3: Perform all multiplication and division in order from left to right.

Step 4: Perform all addition and subtraction in order from left to right.

To help you remember the order of operations, you can remember the acronym BEDMAS.

B – brackets

E – exponents (and square roots)

DM – division and multiplication

AS – addition and subtraction
Example 1

Simplify {3}^{2}-18\div \left(11-5\right).

Table 1.2.3
Steps Solution
Simplify all expressions inside brackets. {3}^{2}-18\div \left(11-5\right)
{3}^{2}-18\div 6
Simplify all expressions with exponents (and square roots). 9-18\div 6
Perform all multiplication and division in order from left to right. 9-3
Perform all addition and subtraction in order from left to right. 6
Practice 1

Simplify.

a) \phantom{\rule{0.2em}{0ex}}2+6 \bullet 3\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.4em}{0ex}}

 

 

 

 b) \phantom{\rule{0.2em}{0ex}}\text{(2+6)} \bullet \text{3}

 

 

 

 c) 4 \bullet 7+3 \bullet 5

 

 

 
d) 4+6\left(3+6\right)

 

 

 

 

 e) \left(9+12\right)\div \left(3+4\right)

 

 

 

 

 f) 33\div 3+8 \bullet 2

 

 

 

 
g) 55\div \sqrt{121} \bullet 5

 

 

 

 

 

 h) 5\left(2+8 \bullet 4\right)-{7}^{2}

 

 

 

 

 
i) 4\left(7+3\right)\div\sqrt{4}-5 \bullet 6

 

 

 

 

 

 j) 5\left[2+4\left(3-2\right)\right]

 

 

 

 

 

Homework 

  1. Simplify.

    a) \phantom{\rule{0.2em}{0ex}}3+8 \bullet 5\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.4em}{0ex}}

     

     

     

    		 b) \phantom{\rule{0.2em}{0ex}}\text{(3+8)} \bullet \text{5}

     

     

     

    		 c) {2}^{3}-12 \div \left(9-5\right)

     

     

     
    d) 3 \bullet 8+5 \bullet 2

     

     

     

    		 e) 2+8\left(6+1\right)

     

     

     

    		 f) 4 \bullet 12/8

     

     

     
    g) 6+10/2+2

     

     

     

    		 h) \left(6+10\right)\div \left(2+2\right)

     

     

     
    i) 20\div 4+6 \bullet 5

     

     

     

    		 j) 20\div \left(4+6\right) \bullet 5

     

     

     
    k) {4}^{2}+{5}^{2}

     

     

    		 l) {\left(4+5\right)}^{2}

     

     
    m) 3\left(1+9 \bullet 6\right)-{4}^{2}

     

     

     

    		 n) 2\left[1+3\left(10-2\right)\right]

     

     

     
    o) 5 + 3 \times 2

     

     

    		 p) 12-6 \div 3

     

     
    q) \left(24-4\right) \div 5

     

     

    		 r) 6 +\left(9-2\right)^{2}

     

     
    s) \left(22-18\right)\times \left(12-6\right)

     

     

     

    		 t) \left(24+8\right)\div \left(14-10\right)

     

     

     
    u) 78-8 \bullet 9

     

     

     

     

    		 v) 67+\left(8-7\rigth)\times3+5

     

     

     

     
    w) \left(34-14\right)+16\div2\left(10 - 5\right)

     

     

     

     

     

    		 x) 67-2\left(17 + 8\right) - 10 + 2\times4

     

     

     

     

     
    y) \left(5\times7\right)+\left(12-6\right)

     

     

     

     

    		 z) 17 + 12-8\left(12 - 9\right)

     

     

     

     
    aa) 12+\left(2 \bullet 8\right)-18\div2+13

     

     

     

     

     

    		 bb) 64\div\left(17-9\right)+ 3\left(8 + 5\right)-7

     

     

     

     

     

  2. Simplify.

    a) 6\sqrt{25}- 4\sqrt{16}

     

     

    		 b) 5-5^2

     

     

    		 c) 2-(-5)+ 3^2

     

     
    d) -17 + 8\sqrt{9} - \left(11- 5\right)^2

     

     

     

     

     

    		 e) 4(3)^2 +7 \left(3+9\right)-(-6)

     

     

     

     

     

  3. Which of the following is the correct order of operations to solve the expression 3+4\left(2+1\right)^2\div3?

    a) Add first, then multiply, divide, and subtract
    b) Brackets first, then exponents, multiplication and division from left to right, and finally addition
    c) Exponents first, then brackets, multiplication, and division from left to right, and finally addition
    d) Multiplication first, then brackets, exponents, and addition

  4. What would be the first step to evaluate 3+4\left(2+1\right)^2\div3?

    a) Add 3 + 4
    b) Multiply 4 by 3
    c) Square the number 2
    d) Add 2 + 1

  5. When should you simplify square roots within an expression?

    a) After performing all addition and subtraction operations
    b) Only after all multiplication and division have been completed
    c) After simplifying expressions inside the brackets and before performing multiplication and division
    d) Before solving brackets and exponents

  6. Which of the following statements correctly describes the order of operations according to BEDMAS?

    a) Operations inside brackets come last, and multiplication and division are performed before addition and subtraction.
    b) Exponents should be applied before parentheses, and division should always come before multiplication.
    c) Operations inside brackets are performed first, followed by exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).
    d) Addition and subtraction are performed first, followed by multiplication and division, then exponents and parentheses.

Answers

1.

a) 43 b) 55 c) 5 d) 34 e) 58 f) 6 g) 13
h) 4 i) 35 j) 10 k) 41 l) 81 m) 149 n) 50
o) 11 p) 10 q) 4 r) 55 s) 24 t) 8 u) 6
v) 75 w) 60 x) 15 y) 41 z) 5 aa) 32 bb) 40

2.

a) 14 b) –20 c) 16 d) –29 e) 126
3. b 4. d 5. c 6. c

 

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resource:

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

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