How to use order of operations

How to Use Order of Operations

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The order of operations is the agreed order for simplifying expressions that contain more than one operation. It keeps everyone from getting different answers to the same problem.

A helpful memory device is PEMDAS: parentheses, exponents, multiplication and division, addition and subtraction. Multiplication and division have the same priority, so work them from left to right. Addition and subtraction also have the same priority, so work them from left to right.

Parentheses and grouping symbols: Simplify inside parentheses, brackets, braces, fraction bars, or absolute value bars first.
Exponents: Evaluate powers after grouping symbols are simplified.
Multiplication and division: Work left to right.
Addition and subtraction: Work left to right.

Example 1

Evaluate \(7 + 3(8 - 5)^2\). First simplify the parentheses: \(8 - 5 = 3\). Then evaluate the exponent: \(3^2 = 9\). Multiply: \(3 \times 9 = 27\). Add: \(7 + 27 = 34\).

Example 2

Evaluate \(24 \div 3 \times 2\). Division and multiplication are tied, so move left to right: \(24 \div 3 = 8\), then \(8 \times 2 = 16\). Do not multiply \(3 \times 2\) first just because multiplication appears in PEMDAS before division in the word.

Order of Operations

Think of this lesson as more than a rule to memorize. Order of Operations is about number-line meaning, signs, and distance from zero. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

A ratio compares quantities, and a proportion says two ratios are equal. Cross products help because \(\frac{a}{b}=\frac{c}{d}\) implies \(ad=bc\).

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

Read what is given and what is being asked.
Choose the rule that connects them.
Substitute carefully and simplify in small steps.
Check the final answer against the original question.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Order of Operations

1) \(6 + 3 \times 4 =\)

2) \((18 - 6) \div 3 =\)

3) \(5^2 - 7 =\)

4) \(30 \div 5 + 2^3 =\)

5) \(4(9 - 3) + 2 =\)

6) \(7 + 18 \div 3 \times 2 =\)

7) \(48 \div (2^3 - 2) =\)

8) \(3^2 + 4(10 - 6) =\)

9) \(60 - 5[3 + 2(4)] =\)

10) \(2^4 \div 4 + 6(3 - 1) =\)

11) \(45 \div 3^2 + (8 - 5)^3 =\)

12) \(6 + 2[15 - (4^2 - 7)] =\)

13) \((36 \div 4)^2 - 5(7 + 1) =\)

14) \(100 - [18 \div 3 + 4(6 - 2)] =\)

15) \(3\{12 - [2^3 + (9 - 6)]\} =\)

16) \(72 \div [3(2 + 4)] + 5^2 =\)

17) \(2^3(15 - 9) \div 4 + 7 =\)

18) \([5^2 - 3(4 + 2)]^2 \div 7 =\)

19) \(120 \div \{5[2^3 - (6 - 4)]\} + 9 =\)

20) \(4^3 - 2\{18 \div [3 + (5 - 2)]\} + 7 =\)

Answers

1) \(6 + 3 \times 4\). Multiply first: \(3 \times 4 = 12\). Then add: \(6 + 12 = \color{red}{18}\).

2) \((18 - 6) \div 3\). Parentheses first: \(18 - 6 = 12\). Then divide: \(12 \div 3 = \color{red}{4}\).

3) \(5^2 - 7\). Exponent first: \(5^2 = 25\). Then subtract: \(25 - 7 = \color{red}{18}\).

4) \(30 \div 5 + 2^3\). Exponent: \(2^3 = 8\). Divide: \(30 \div 5 = 6\). Add: \(6 + 8 = \color{red}{14}\).

5) \(4(9 - 3) + 2\). Parentheses: \(9 - 3 = 6\). Multiply: \(4 \times 6 = 24\). Add: \(24 + 2 = \color{red}{26}\).

6) \(7 + 18 \div 3 \times 2\). Divide and multiply left to right: \(18 \div 3 = 6\), then \(6 \times 2 = 12\). Add: \(7 + 12 = \color{red}{19}\).

7) \(48 \div (2^3 - 2)\). Inside parentheses, exponent first: \(2^3 = 8\). Then \(8 - 2 = 6\). Divide: \(48 \div 6 = \color{red}{8}\).

8) \(3^2 + 4(10 - 6)\). Parentheses: \(10 - 6 = 4\). Exponent: \(3^2 = 9\). Multiply: \(4 \times 4 = 16\). Add: \(9 + 16 = \color{red}{25}\).

9) \(60 - 5[3 + 2(4)]\). Brackets first: \(2(4)=8\), so \(3 + 8 = 11\). Multiply: \(5 \times 11 = 55\). Subtract: \(60 - 55 = \color{red}{5}\).

10) \(2^4 \div 4 + 6(3 - 1)\). Parentheses: \(3 - 1 = 2\). Exponent: \(2^4 = 16\). Divide: \(16 \div 4 = 4\). Multiply: \(6 \times 2 = 12\). Add: \(4 + 12 = \color{red}{16}\).

11) \(45 \div 3^2 + (8 - 5)^3\). Parentheses: \(8 - 5 = 3\). Exponents: \(3^2=9\) and \(3^3=27\). Divide: \(45 \div 9 = 5\). Add: \(5 + 27 = \color{red}{32}\).

12) \(6 + 2[15 - (4^2 - 7)]\). Inner parentheses: \(4^2 - 7 = 16 - 7 = 9\). Brackets: \(15 - 9 = 6\). Multiply: \(2 \times 6 = 12\). Add: \(6 + 12 = \color{red}{18}\).

13) \((36 \div 4)^2 - 5(7 + 1)\). Parentheses: \(36 \div 4 = 9\) and \(7+1=8\). Exponent: \(9^2 = 81\). Multiply: \(5 \times 8 = 40\). Subtract: \(81 - 40 = \color{red}{41}\).

14) \(100 - [18 \div 3 + 4(6 - 2)]\). Parentheses: \(6 - 2 = 4\). Divide: \(18 \div 3 = 6\). Multiply: \(4 \times 4 = 16\). Brackets: \(6 + 16 = 22\). Subtract: \(100 - 22 = \color{red}{78}\).

15) \(3\{12 - [2^3 + (9 - 6)]\}\). Parentheses: \(9 - 6 = 3\). Exponent: \(2^3 = 8\). Brackets: \(8 + 3 = 11\). Braces: \(12 - 11 = 1\). Multiply: \(3 \times 1 = \color{red}{3}\).

16) \(72 \div [3(2 + 4)] + 5^2\). Parentheses: \(2 + 4 = 6\). Brackets: \(3 \times 6 = 18\). Exponent: \(5^2 = 25\). Divide: \(72 \div 18 = 4\). Add: \(4 + 25 = \color{red}{29}\).

17) \(2^3(15 - 9) \div 4 + 7\). Parentheses: \(15 - 9 = 6\). Exponent: \(2^3 = 8\). Multiply and divide left to right: \(8 \times 6 = 48\), then \(48 \div 4 = 12\). Add: \(12 + 7 = \color{red}{19}\).

18) \([5^2 - 3(4 + 2)]^2 \div 7\). Parentheses: \(4 + 2 = 6\). Exponent: \(5^2 = 25\). Multiply: \(3 \times 6 = 18\). Brackets: \(25 - 18 = 7\). Square: \(7^2 = 49\). Divide: \(49 \div 7 = \color{red}{7}\).

19) \(120 \div \{5[2^3 - (6 - 4)]\} + 9\). Parentheses: \(6 - 4 = 2\). Exponent: \(2^3 = 8\). Brackets: \(8 - 2 = 6\). Braces: \(5 \times 6 = 30\). Divide: \(120 \div 30 = 4\). Add: \(4 + 9 = \color{red}{13}\).

20) \(4^3 - 2\{18 \div [3 + (5 - 2)]\} + 7\). Parentheses: \(5 - 2 = 3\). Brackets: \(3 + 3 = 6\). Exponent: \(4^3 = 64\). Inside braces: \(18 \div 6 = 3\). Multiply: \(2 \times 3 = 6\). Then \(64 - 6 + 7 = \color{red}{65}\).

How to use order of operations