Operation Hierarchy Examples: 48 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Operation Hierarchy.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The layered relationship between arithmetic operations, where each is built from the previous: multiplication from addition, exponentiation from multiplication.

Multiplication is repeated addition. Exponents are repeated multiplication.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Multiplication is repeated addition, and exponentiation is repeated multiplication — a ladder of operations.

Common stuck point: The procedure for operation hierarchy is the easy part; the trap is treating exponentiation as repeated addition. Asking "Am I describing one operation as the repetition of a simpler one?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I describing one operation as the repetition of a simpler one?

Worked Examples

Example 1

easy

Evaluate

3 + 4 \times 2

using the correct order of operations.

Answer

First step

Order of operations (PEMDAS): Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

Full solution

2
No parentheses or exponents.
3
Multiplication first: $4 \times 2 = 8$ .
4
Then addition: $3 + 8 = 11$ .

Multiplication is performed before addition in the order of operations.

challenge

Evaluate

2^{2^3}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Evaluate

10 - 2 \times 3 + 1

Example 2

medium

Evaluate

(5 + 3)^2 \div 4 - 6

Example 3

easy

2 + 3 \times 4

, which operation is performed first?

Example 4

easy

Evaluate

2 + 3 \times 4

Example 5

easy

Rewrite

5 \times 3

as repeated addition.

Example 6

easy

Rewrite

2^3

as repeated multiplication.

Example 7

easy

8 - 6 \div 2

, which operation comes first?

Example 8

easy

Evaluate

10 - 2 \times 3

Example 9

easy

2^3

equal to

2 \times 3

? (yes/no)

Example 10

easy

Evaluate

3 + 4^2

Example 11

medium

Evaluate

2 + 3 \times 4^2

Example 12

medium

Evaluate

(2 + 3) \times 4^2

Example 13

medium

Evaluate

20 \div 4 \times 5

and explain the order.

Example 14

medium

Evaluate

36 \div 6 + 2 \times 3

Example 15

medium

Explain why

4 \times 5

can be seen as one tier above

5 + 5 + 5 + 5

Example 16

medium

Evaluate

2 \times 3^2 - 4

Example 17

medium

Why is exponentiation considered 'higher' than multiplication in the hierarchy?

Example 18

medium

Evaluate

18 - 2^3 + 5

Example 19

medium

Evaluate

40 \div 8 + 2^2 \times 3

Example 20

challenge

Evaluate

2 + 3 \times (4 - 1)^2

Example 21

challenge

Insert parentheses into

1 + 2 \times 3 + 4

to make the value equal 21. Show the grouping.

Example 22

challenge

Evaluate

100 \div 5^2 \times 2 + 3

Example 23

easy

Evaluate

5 + 6 \div 3

Example 24

easy

Evaluate

9 - 2^2

Example 25

easy

Rewrite

4^2

4 + 6 \div 2 \times 3

so the value equals

5

Example 41

challenge

Evaluate

\left( \dfrac{2^3 + 4}{3} \right) \times (5 - 2^2)

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Related Concepts

Addition Multiplication Exponents

Background Knowledge

These ideas may be useful before you work through the harder examples.

additionmultiplicationexponents