The rules for solving mathematical expressions give you a way to decide the order in which an expression gets evaluated. This set of rules is called the order of operations (or sometimes, the order of precedence). Here’s the complete order of operations for arithmetic:

1. Contents of parentheses from the inside out

2. Powers from left to right

3. Multiplication and division from left to right

4. Addition and subtraction from left to right

## Sample question

1. Evaluate [(8 x 4 + 23) / 10]7–5.

16. Start by focusing on the inner set of parentheses, evaluating the power, then the multiplication, and then the addition:

[(8 x 4 + 23) / 10]7–5

= [(8 x 4 + 8) / 10]7–5

= [(32 + 8) / 10]7–5

= [40 / 10]7–5

Next, evaluate what’s inside the parentheses and the expression that makes up the exponent:

= 47–5 = 42

Finish by evaluating the remaining power: 42 = 16.

## Practice questions

1. Evaluate 1 + [(23 – 4) + (10 / 2)2].

2. (–7 x –2 + 62 / 4)9x2–17

3. What is {62 – [12 / (–13 + 14)2] x 2}2?

4. Find the value of [(123 – 112)4 – (62 / 220–3x6)]2.

Following are the answers to the practice questions:

1. 1 + [(23 – 4) + (10 / 2)2] = 30.

Start by focusing on the first of the two inner sets of parentheses, (23 – 4). Evaluate the power first and then the subtraction:

1 + [(23 – 4) + (10 / 2)2] = 1 + [(8 – 4) + (10 / 2)2] = 1 + [4 + (10 / 2)2]

Continue by focusing on the remaining inner set of parentheses:

= 1 + [4 + 52]

Next, evaluate what’s inside the last set of parentheses, evaluating the power first and then the addition:

= 1 + [4 + 25] = 1 + 29

Finish by adding the remaining numbers:

1 + 29 = 30

2. (–7 x –2 + 62 / 4)9x2–17 = 23.

Start with the first set of parentheses. Evaluate the power first, then the multiplication and division from left to right, and then the addition:

(–7 x –2 + 62 / 4)9x2–17

= (–7 x –2 + 36 / 4)9x2–17

= (14 + 36 / 4)9x2–17

= (14 + 9)9x2–17

= 239x2–17

Next, work on the exponent, evaluating the multiplication first and then the subtraction:

= 2318–17 = 231

Finish by evaluating the power:

231 = 23

3. {62 – [12 / (–13 + 14)2] x 2}2 = 144.

Start by evaluating the inner set of parentheses (–13 + 14):

{62 – [12 / (–13 + 14)2] x 2}2

= {62 – [12 / 12] x 2}2

Move outward to the next set of parentheses, [12 / 12], evaluating the power and then the division:

= {62 – [12 / 1] x 2}2

= {62 – 12 x 2}2

Next, work on the remaining set of parentheses, evaluating the power, then the multiplication, and then the subtraction:

= {36 – 12 x 2}2

= {36 – 24}2

= 122

Finish by evaluating the power:

122 = 144

4. [(123 – 11)2)4 – (62 / 220–3x6)]2 = 49.

Start by working on the exponent, 20 – 3 x 6, evaluating the multiplication and then the subtraction:

[(123 – 11)2)4 – (62 / 220–3x6)]2

= [(123 – 11)2)4 – (62 / 220–18)]2

= [(123 – 11)2)4 – (62 / 22)]2

The result is an expression with two inner sets of parentheses. Focus on the first of these two sets, evaluating the power and then the subtraction:

= [(123 – 121)4 – (62 / 22)]2

Work on the remaining inner set of parentheses, evaluating the two powers from left to right and then the division:

= [24 – (36 / 22)]2

= [24 – (36 / 4)]2

= [24 – 9]2

Now evaluate what’s left inside the parentheses, evaluating the power and then the subtraction:

= [16 – 9]2

= 72

Finish by evaluating the power: 72 = 49.