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: Operations form a hierarchy of increasingly powerful repeated actions.

Common stuck point: The hierarchy breaks down with non-integers (how to repeat 2.5 times?).

Sense of Study hint: Write out the chain: show how the higher operation expands into repeated use of the lower one.

Worked Examples

Example 1

easy
Evaluate \(3 + 4 \times 2\) using the correct order of operations.

Solution

  1. 1
    Order of operations (PEMDAS): Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
  2. 2
    No parentheses or exponents.
  3. 3
    Multiplication first: \(4 \times 2 = 8\).
  4. 4
    Then addition: \(3 + 8 = 11\).

Answer

11
Multiplication is performed before addition in the order of operations. \(3 + 4 \times 2 = 3 + 8 = 11\), not \((3+4) \times 2 = 14\).

Example 2

medium
Evaluate \(2 + 3^2 \times (4 - 1) \div 9\).

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 \((5 + 3)^2 \div 4 - 6\).
← Back to Operation Hierarchy Formula Explained Practice Mode

Background Knowledge

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

additionmultiplicationexponents