Math Vale

🔢 Number Kingdom · Number Sense

Order of Operations

Evaluate expressions in the agreed order: parentheses, exponents, multiplication and division left to right, then addition and subtraction left to right.

In short

  • Parentheses, then exponents, then multiplication and division, then addition and subtraction.
  • Multiplication and division share one rank, as do addition and subtraction: at equal rank, go left to right.
  • An exponent applies only to the base immediately below it, unless brackets say otherwise.
  • Fraction bars and root signs act as invisible brackets.

Why an order is needed at all

Consider 2 + 3 x 4. Working left to right gives 20. Multiplying first gives 14. Both are defensible readings — so mathematicians agreed on one, and every calculator, spreadsheet and programming language now follows it.

The agreed order is:

1. Parentheses (brackets), innermost first 2. Exponents (powers and roots) 3. Multiplication and division, left to right 4. Addition and subtraction, left to right

So 2 + 3 x 4 = 2 + 12 = 14. This is a convention, not a discovery, but it is universal: without it, an expression would not have a single meaning.

Same rank means left to right

Steps 3 and 4 each contain two operations of equal rank, and neither one beats the other. Where they meet, work strictly left to right.

  • 20 - 8 - 5 = 12 - 5 = 7 (not 20 - 3 = 17)
  • 48 / 6 / 2 = 8 / 2 = 4 (not 48 / 3 = 16)
  • 30 / 5 x 2 = 6 x 2 = 12 (not 30 / 10 = 3)

The letters in "PEMDAS" or "BIDMAS" hide this: MD and AS are single steps read left to right, not four separate stages.

Parentheses take priority

Whatever is inside brackets is worked out first, and the ordinary rules apply inside them too.

  • 5 x (3 + 4) = 5 x 7 = 35, while 5 x 3 + 4 = 19.
  • 3 x (2 + 4 x 5) = 3 x (2 + 20) = 3 x 22 = 66. Inside the brackets, the multiplication still went first.

Brackets are also how you *change* the order deliberately. If you want an addition done before a multiplication, brackets are the only way to say so.

Exponents bind tightly

An exponent attaches only to the thing immediately beneath it.

  • 3 x 42 = 3 x 16 = 48, not 122 = 144.
  • -32 = -(32) = -9, while (-3)2 = 9.
  • 2 x 52 + 1 = 2 x 25 + 1 = 51.

If you mean the whole product to be squared, you must write brackets: (3 x 4)2.

Working cleanly

Long expressions go wrong through bookkeeping, not through misunderstanding. A reliable routine:

  • Copy the whole expression onto a new line after every single step.
  • Change exactly one thing per line.
  • Keep everything else identical, in the same order.

A fraction bar and a square-root sign are also grouping symbols: they act like invisible brackets around everything above, below, or inside them.

Worked examples

Example 1

6 + 4 x (9 - 5)2 / 8 = ?

  1. Parentheses first: 9 - 5 = 4, giving 6 + 4 x 42 / 8.
  2. Exponents next: 42 = 16, giving 6 + 4 x 16 / 8.
  3. Multiplication and division, left to right: 4 x 16 = 64, giving 6 + 64 / 8.
  4. Still left to right: 64 / 8 = 8, giving 6 + 8.
  5. Addition last: 6 + 8 = 14.

Example 2

40 - 12 / 3 - 2 x 5 = ?

  1. No brackets and no exponents, so start with multiplication and division, left to right.
  2. 12 / 3 = 4, giving 40 - 4 - 2 x 5.
  3. 2 x 5 = 10, giving 40 - 4 - 10.
  4. Now addition and subtraction, left to right: 40 - 4 = 36.
  5. 36 - 10 = 26.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

53 - 5 x 4 = ?

Answer: 33

  1. 53 - 5 x 4
  2. Do 5 x 4 first, then subtract that from the first number.
  3. = 33

Problem 2

Difficulty 3 of 5

8 x (77 - 44) = ?

Answer: 264

  1. 8 x (77 - 44)
  2. Work out 77 - 44 inside the brackets before multiplying by 8.
  3. = 264

Problem 3

Difficulty 4 of 5

(2 + 2)2 - 1 = ?

Answer: 15

  1. (2 + 2)2 - 1
  2. The whole bracket is squared, so add inside the brackets before squaring.
  3. = 15

Common mistakes

  • Evaluating strictly left to right and getting 2 + 3 x 4 = 20.
  • Doing all multiplication before all division (or all addition before all subtraction) instead of going left to right.
  • Ignoring brackets, or resolving the outer bracket before the inner one.
  • Applying an exponent to a whole product, writing 3 x 42 as 144.

What you should be able to do

  • Evaluate an expression using the correct order of operations.
  • Explain why 2 + 3 * 4 is 14 and not 20.
  • Handle nested parentheses and a fraction bar as a grouping symbol.
  • Insert parentheses to make an expression equal a target value.

Where this fits in the curriculum

Common Core

  • 5.OA.A.1

    Grade 5 — Use parentheses, brackets and braces in numerical expressions and evaluate them.

  • 6.EE.A.2.C

    Grade 6 — Evaluate expressions, performing the operations in the conventional order, including whole-number exponents.

Ontario

  • G7.B2.1

    Grade 7 — Use the properties and order of operations with whole numbers, decimals, fractions, ratios, rates and percents.

  • G8.B2.1

    Grade 8 — Use the properties and order of operations with rational numbers in expressions and problems.

Learn these first

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