Operation Hierarchy Examples in Math

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+4×23 + 4 \times 2 using the correct order of operations.

Answer

11

First step

1
Order of operations (PEMDAS): Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

Full solution

  1. 2
    No parentheses or exponents.
  2. 3
    Multiplication first: 4×2=84 \times 2 = 8.
  3. 4
    Then addition: 3+8=113 + 8 = 11.
Multiplication is performed before addition in the order of operations. 3+4×2=3+8=113 + 4 \times 2 = 3 + 8 = 11, not (3+4)×2=14(3+4) \times 2 = 14.

Example 2

medium
Evaluate 2+32×(4−1)÷92 + 3^2 \times (4 - 1) \div 9.

Example 3

medium
Evaluate 6+2×5−4÷26 + 2 \times 5 - 4 \div 2.

Example 4

medium
Evaluate (6+24)2\left( \dfrac{6 + 2}{4} \right)^2.

Example 5

hard
Evaluate 16+3×22−6÷3\sqrt{16} + 3 \times 2^2 - 6 \div 3.

Example 6

hard
Evaluate 5−2×[3+(4−2)2]5 - 2 \times [3 + (4 - 2)^2].

Example 7

challenge
Evaluate 2232^{2^3}.

Practice Problems

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

Example 1

easy
Evaluate 10−2×3+110 - 2 \times 3 + 1.

Example 2

medium
Evaluate (5+3)2÷4−6(5 + 3)^2 \div 4 - 6.

Example 3

easy
In 2+3×42 + 3 \times 4, which operation is performed first?

Example 4

easy
Evaluate 2+3×42 + 3 \times 4.

Example 5

easy
Rewrite 5×35 \times 3 as repeated addition.

Example 6

easy
Rewrite 232^3 as repeated multiplication.

Example 7

easy
In 8−6÷28 - 6 \div 2, which operation comes first?

Example 8

easy
Evaluate 10−2×310 - 2 \times 3.

Example 9

easy
Is 232^3 equal to 2×32 \times 3? (yes/no)

Example 10

easy
Evaluate 3+423 + 4^2.

Example 11

medium
Evaluate 2+3×422 + 3 \times 4^2.

Example 12

medium
Evaluate (2+3)×42(2 + 3) \times 4^2.

Example 13

medium
Evaluate 20÷4×520 \div 4 \times 5 and explain the order.

Example 14

medium
Evaluate 36÷6+2×336 \div 6 + 2 \times 3.

Example 15

medium
Explain why 4×54 \times 5 can be seen as one tier above 5+5+5+55 + 5 + 5 + 5.

Example 16

medium
Evaluate 2×32−42 \times 3^2 - 4.

Example 17

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

Example 18

medium
Evaluate 18−23+518 - 2^3 + 5.

Example 19

medium
Evaluate 40÷8+22×340 \div 8 + 2^2 \times 3.

Example 20

challenge
Evaluate 2+3×(4−1)22 + 3 \times (4 - 1)^2.

Example 21

challenge
Insert parentheses into 1+2×3+41 + 2 \times 3 + 4 to make the value equal 21. Show the grouping.

Example 22

challenge
Evaluate 100÷52×2+3100 \div 5^2 \times 2 + 3.

Example 23

easy
Evaluate 5+6÷35 + 6 \div 3.

Example 24

easy
Evaluate 9−229 - 2^2.

Example 25

easy
Rewrite 424^2 as repeated multiplication.

Example 26

easy
Evaluate 12÷4+312 \div 4 + 3.

Example 27

easy
Evaluate 1+2×31 + 2 \times 3.

Example 28

easy
Evaluate (1+2)×3(1 + 2) \times 3.

Example 29

medium
Evaluate 32+423^2 + 4^2.

Example 30

medium
Evaluate 5×235 \times 2^3.

Example 31

medium
Evaluate 42−15\dfrac{4^2 - 1}{5}.

Example 32

medium
Evaluate 50−32×250 - 3^2 \times 2.

Example 33

medium
Evaluate 23+222^3 + 2^2.

Example 34

medium
Evaluate 48÷2÷448 \div 2 \div 4.

Example 35

medium
Evaluate 10−2×(3+4÷2)10 - 2 \times (3 + 4 \div 2).

Example 36

hard
Evaluate 21+22^{1+2}.

Example 37

hard
Evaluate 3+2×42−10÷53 + 2 \times 4^2 - 10 \div 5.

Example 38

hard
Evaluate 24−2×35\dfrac{2^4 - 2 \times 3}{5}.

Example 39

hard
Evaluate −32+(−3)2-3^2 + (-3)^2.

Example 40

challenge
Insert one pair of parentheses into 4+6÷2×34 + 6 \div 2 \times 3 so the value equals 55.

Example 41

challenge
Evaluate (23+43)×(5−22)\left( \dfrac{2^3 + 4}{3} \right) \times (5 - 2^2).
← Back to Operation Hierarchy Formula Explained Practice Mode

Background Knowledge

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

additionmultiplicationexponents