# Practice Solving Expressions with the Order of Operations

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:

Contents of parentheses from the inside out

Powers from left to right

Multiplication and division from left to right

Addition and subtraction from left to right

## Sample question

Evaluate [(8 x 4 + 2

^{3}) / 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 + 2

^{3}) / 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:

= 4

^{7–5}= 4^{2}^{}Finish by evaluating the remaining power: 4

^{2}= 16.

## Practice questions

Evaluate 1 + [(2

^{3}– 4) + (10 / 2)^{2}].(–7 x –2 + 6

^{2}/ 4)^{9x2–17}^{}What is {6

^{2}– [12 / (–13 + 14)^{2}] x 2}^{2}?Find the value of [(123 – 11

^{2})^{4}– (6^{2}/ 2^{20–3x6})]^{2}.

Following are the answers to the practice questions:

1 + [(2

^{3}– 4) + (10 / 2)^{2}] =**30.**Start by focusing on the first of the two inner sets of parentheses, (2

^{3}– 4). Evaluate the power first and then the subtraction:1 + [(2

^{3}– 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 + 5

^{2}]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

(–7 x –2 + 6

^{2}/ 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 + 6

^{2}/ 4)^{9x2–17}= (–7 x –2 + 36 / 4)

^{9x2–17}= (14 + 36 / 4)

^{9x2–17}= (14 + 9)

^{9x2–17}= 23

^{9x2–17}Next, work on the exponent, evaluating the multiplication first and then the subtraction:

= 23

^{18–17}= 23^{1}Finish by evaluating the power:

23

^{1}= 23{6

^{2}– [12 / (–13 + 14)^{2}] x 2}^{2}= 144.Start by evaluating the inner set of parentheses (–13 + 14):

{6

^{2}– [12 / (–13 + 14)^{2}] x 2}^{2}= {6

^{2}– [12 / 1^{2}] x 2}^{2}Move outward to the next set of parentheses, [12 / 1

^{2}], evaluating the power and then the division:= {6

^{2}– [12 / 1] x 2}^{2}= {6

^{2}– 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}= 12

^{2}Finish by evaluating the power:

12

^{2}= 144[(123 – 11)

^{2})^{4}– (6^{2}/ 2^{20–3x6})]^{2}= 49.Start by working on the exponent, 20 – 3 x 6, evaluating the multiplication and then the subtraction:

[(123 – 11)

^{2})^{4}– (6^{2}/ 2^{20–3x6})]^{2}= [(123 – 11)

^{2})^{4}– (6^{2}/ 2^{20–18})]^{2}= [(123 – 11)

^{2})^{4}– (6^{2}/ 2^{2})]^{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}– (6^{2}/ 2^{2})]^{2}^{}Work on the remaining inner set of parentheses, evaluating the two powers from left to right and then the division:

= [2

^{4}– (36 / 2^{2})]^{2}= [2

^{4}– (36 / 4)]^{2}= [2

^{4}– 9]^{2}Now evaluate what’s left inside the parentheses, evaluating the power and then the subtraction:

= [16 – 9]

^{2}= 7

^{2}Finish by evaluating the power: 7

^{2}= 49.