Order of Operations

Mathematical expressions often contain several different operations. If everyone performed those operations in whatever order they wanted, the same expression could produce different answers.

The order of operations is a set of agreed mathematical rules that tells us which calculations should be completed first. It ensures that a mathematical expression has a consistent meaning.

Two common ways of remembering the order are PEMDAS and BEDMAS.

Learning Targets

• I can explain the purpose of the order of operations.
• I can apply PEMDAS/BEDMAS correctly.
• I can evaluate numerical expressions involving multiple operations.
• I can evaluate algebraic expressions by substituting values.
• I can solve multi-step calculations accurately.

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Why Do We Need an Order of Operations?

Consider:

3 + 4 × 2

There are at least two operations:

• Addition
• Multiplication

If we simply work from left to right:

3 + 4 = 7

7 × 2 = 14

But if we multiply first:

4 × 2 = 8

3 + 8 = 11

We have two different answers.

Mathematics therefore needs an agreed set of rules.

Using the correct order of operations:

3 + 4 × 2

= 3 + 8

= 11

The correct answer is:

11

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PEMDAS and BEDMAS

Two commonly used abbreviations are:

PEMDAS

and:

BEDMAS

They describe essentially the same rules.

A more accurate way to think about the order is:

Brackets or Parentheses

then:

Exponents

then:

Multiplication and Division from left to right

then:

Addition and Subtraction from left to right

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Multiplication and Division Have Equal Priority

A common misunderstanding is that multiplication must always happen before division because the M appears before the D in PEMDAS.

That is incorrect.

Multiplication and division have equal priority.

When both appear, work from:

Left to right

Consider:

24 ÷ 6 × 2

Start from the left:

24 ÷ 6 = 4

Then:

4 × 2 = 8

Therefore:

24 ÷ 6 × 2 = 8

Do not automatically perform multiplication before division.

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Addition and Subtraction Have Equal Priority

The same rule applies to addition and subtraction.

They have equal priority.

Work from:

Left to right

Consider:

10 − 4 + 3

Start from the left:

10 − 4 = 6

Then:

6 + 3 = 9

Therefore:

10 − 4 + 3 = 9

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Brackets and Parentheses

Operations inside brackets or parentheses are normally completed first.

Consider:

3 × (4 + 2)

Calculate inside the parentheses:

4 + 2 = 6

Then:

3 × 6 = 18

Therefore:

3 × (4 + 2) = 18

Compare this with:

3 × 4 + 2

Without brackets:

3 × 4 = 12

12 + 2 = 14

So:

3 × (4 + 2) = 18

but:

3 × 4 + 2 = 14

Brackets can completely change the meaning of an expression.

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Exponents

After dealing with brackets, calculate exponents.

An exponent tells us how many times a number is multiplied by itself.

For example:

4² = 4 × 4 = 16

and:

2³ = 2 × 2 × 2 = 8

Consider:

5 + 3²

Calculate the exponent first:

3² = 9

Then:

5 + 9 = 14

Therefore:

5 + 3² = 14

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Worked Example — Basic BEDMAS

Evaluate:

8 + 3 × 4

There are no brackets or exponents.

Multiplication comes before addition.

3 × 4 = 12

Then:

8 + 12 = 20

Therefore:

8 + 3 × 4 = 20

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Worked Example — Including Subtraction

Evaluate:

20 − 4 × 3

Multiplication first:

4 × 3 = 12

Then subtract:

20 − 12 = 8

Therefore:

20 − 4 × 3 = 8

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Worked Example — Including Division

Evaluate:

18 ÷ 3 + 5

Division comes before addition:

18 ÷ 3 = 6

Then:

6 + 5 = 11

Therefore:

18 ÷ 3 + 5 = 11

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Worked Example — Brackets

Evaluate:

5 × (7 − 3)

Brackets first:

7 − 3 = 4

Then:

5 × 4 = 20

Therefore:

5 × (7 − 3) = 20

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Worked Example — Exponents

Evaluate:

2 + 3² × 4

First calculate the exponent:

3² = 9

Now:

2 + 9 × 4

Multiplication:

9 × 4 = 36

Finally:

2 + 36 = 38

Therefore:

2 + 3² × 4 = 38

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Multiple Operations

More complicated expressions may require several stages.

Consider:

18 − 2 × (3 + 4)

Start with the brackets:

3 + 4 = 7

Now:

18 − 2 × 7

Multiplication:

2 × 7 = 14

Finally:

18 − 14 = 4

Therefore:

18 − 2 × (3 + 4) = 4

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Nested Brackets

Sometimes brackets appear inside other brackets.

For example:

2[5 + (8 − 3)]

Begin with the innermost brackets:

8 − 3 = 5

Now:

2[5 + 5]

Calculate the remaining brackets:

5 + 5 = 10

Then:

2 × 10 = 20

Therefore:

2[5 + (8 − 3)] = 20

When brackets are nested, work from the inside outward.

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A Reliable Strategy

For complicated expressions, use the following process:

• Look for brackets.
• Calculate exponents.
• Complete multiplication and division from left to right.
• Complete addition and subtraction from left to right.
• Work one stage at a time.
• Rewrite the expression after each stage.

Writing each stage greatly reduces mistakes.

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Evaluating Algebraic Expressions

The order of operations is also important when evaluating algebraic expressions.

Consider:

3x + 5

If:

x = 4

substitute 4 for x:

3(4) + 5

Multiplication first:

3 × 4 = 12

Then:

12 + 5 = 17

Therefore:

3x + 5 = 17 when x = 4

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Substitution

Substitution means replacing a variable with its known value.

Suppose:

x = 5

Evaluate:

2x + 7

Substitute:

2(5) + 7

Multiply:

10 + 7

Therefore:

17

Using parentheses around substituted values is often helpful because it makes the multiplication clear.

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Worked Example — Substitution with Exponents

If:

x = 3

evaluate:

x² + 4x

Substitute 3:

(3)² + 4(3)

Calculate the exponent:

9 + 4(3)

Multiply:

9 + 12

Add:

21

Therefore:

x² + 4x = 21 when x = 3

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Worked Example — Two Variables

If:

a = 4

and:

b = 3

evaluate:

2a + 5b

Substitute:

2(4) + 5(3)

Multiply:

8 + 15

Add:

23

Therefore:

2a + 5b = 23

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Substitution with Brackets

Suppose:

x = 3

Evaluate:

2(x + 5)

Substitute:

2(3 + 5)

Brackets first:

2(8)

Multiply:

16

Therefore:

2(x + 5) = 16

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Substitution with Negative Numbers

Extra care is required when substituting negative numbers.

Suppose:

x = −3

Evaluate:

x² + 2x

Substitute using parentheses:

(−3)² + 2(−3)

Calculate the exponent:

9 + 2(−3)

Multiply:

9 − 6

Therefore:

3

Using parentheses around the negative number makes the calculation much clearer.

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Why Parentheses Matter with Negative Numbers

Compare:

(−3)²

and:

−3²

These do not mean exactly the same thing.

For:

(−3)²

the entire negative number is squared:

(−3)(−3) = 9

But:

−3²

means:

−(3²)

which gives:

−9

Parentheses can therefore be extremely important.

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Fractions and the Order of Operations

Fractions can also contain several operations.

A fraction bar acts like a grouping symbol.

For example:

(8 + 4) / (5 − 2)

Calculate the numerator:

8 + 4 = 12

Calculate the denominator:

5 − 2 = 3

Then:

12 / 3 = 4

Therefore:

(8 + 4) / (5 − 2) = 4

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Multi-Step Calculation

Evaluate:

24 ÷ (2 + 4) + 3²

First calculate the brackets:

2 + 4 = 6

Now:

24 ÷ 6 + 3²

Calculate the exponent:

3² = 9

Now:

24 ÷ 6 + 9

Division:

24 ÷ 6 = 4

Finally:

4 + 9 = 13

Therefore:

24 ÷ (2 + 4) + 3² = 13

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Another Multi-Step Example

Evaluate:

5 + 2 × (7 − 3)²

Brackets first:

7 − 3 = 4

Now:

5 + 2 × 4²

Exponent:

4² = 16

Now:

5 + 2 × 16

Multiplication:

2 × 16 = 32

Finally:

5 + 32 = 37

Therefore:

5 + 2 × (7 − 3)² = 37

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Using a Calculator

Scientific calculators are designed to follow the order of operations.

For example, entering:

3 + 4 × 2

should produce:

11

because the calculator performs multiplication before addition.

However, calculators are only useful if expressions are entered correctly.

Brackets are especially important.

Entering:

3 × (4 + 2)

is different from entering:

3 × 4 + 2

The calculator follows what you enter, not what you intended to enter.

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Checking Whether an Answer Is Reasonable

After completing a multi-step calculation, check your answer.

Ask:

• Did I calculate the brackets first?
• Did I calculate exponents correctly?
• Did I perform multiplication and division from left to right?
• Did I perform addition and subtraction from left to right?
• Did I copy each number and operation correctly?
• Did I substitute the correct values?
• Did I handle negative numbers correctly?

Checking your work is particularly important for longer calculations.

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Order of Operations in Science

The order of operations is important beyond mathematics.

Scientific equations often involve several calculations.

For example:

KE = 1/2 mv²

Suppose:

m = 4 kg

and:

v = 3 m/s

Substitute:

KE = 1/2 × 4 × 3²

Exponent first:

3² = 9

Then:

KE = 1/2 × 4 × 9

Therefore:

KE = 18 J

Correct order of operations is essential when using formulas in physics, chemistry, engineering, finance, and computing.

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Order of Operations in Geometry

Suppose a rectangle has:

Length = 8 cm

Width = 5 cm

Its perimeter is:

P = 2(l + w)

Substitute:

P = 2(8 + 5)

Brackets first:

P = 2(13)

Then:

P = 26 cm

This demonstrates why brackets are useful when formulas contain groups of quantities.

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Order of Operations in Everyday Calculations

Imagine buying:

• 3 notebooks at $4 each
• 2 pens at $2 each

The total cost can be written:

3 × 4 + 2 × 2

Perform multiplication first:

12 + 4

Therefore:

$16

The order of operations allows a single mathematical expression to describe the complete calculation.

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Common Mistakes

Mistake: Always working from left to right

Consider:

5 + 3 × 2

Do not calculate:

5 + 3 = 8

then:

8 × 2 = 16

Multiplication comes first:

3 × 2 = 6

then:

5 + 6 = 11

Mistake: Thinking multiplication always comes before division

Multiplication and division have equal priority.

Work from left to right.

Mistake: Thinking addition always comes before subtraction

Addition and subtraction have equal priority.

Work from left to right.

Mistake: Ignoring brackets

2(3 + 4)

does not equal:

2 × 3 + 4

The bracket changes which operation is performed first.

Mistake: Forgetting exponents

In:

4 + 3²

calculate:

3²

before adding 4.

Mistake: Substituting negative values without parentheses

If:

x = −4

write:

(−4)²

rather than immediately writing:

−4²

These have different meanings.

Mistake: Trying to do too many steps mentally

For complicated expressions, write one step at a time.

This makes mistakes easier to identify.

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Key Terms

Order of operations – agreed rules determining the sequence in which mathematical operations are performed.

PEMDAS – Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

BEDMAS – Brackets, Exponents, Division/Multiplication, Addition/Subtraction.

Brackets – grouping symbols indicating calculations that should normally be completed first.

Exponent – a number showing how many times a base is multiplied by itself.

Expression – a mathematical phrase containing numbers, variables, and operations.

Evaluate – calculate the value of an expression.

Substitution – replacing a variable with a known value.

Variable – a symbol representing a number.

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Check Your Understanding

1. Explain why mathematicians use an agreed order of operations.

2. Evaluate:

6 + 4 × 3

3. Evaluate:

20 − 12 ÷ 4

4. Evaluate:

5 × (8 − 3)

5. Evaluate:

3 + 2³

6. Evaluate:

18 ÷ 3 × 2

7. Evaluate:

15 − 6 + 4

8. Evaluate:

4 + 3 × (7 − 5)

9. Evaluate:

2 × (5 + 3)²

10. If:

x = 4

evaluate:

3x + 7

11. If:

x = 5

evaluate:

x² − 3x

12. If:

a = 3

and:

b = 4

evaluate:

2a² + 3b

13. If:

x = −2

evaluate:

x² + 5x

14. Evaluate:

30 ÷ (2 + 3) + 4²

15. Explain why:

(−5)²

and:

−5²

have different values.

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Key Takeaways

• The order of operations ensures that mathematical expressions have consistent meanings.
• PEMDAS and BEDMAS describe the same basic rules.
• Complete brackets or parentheses first.
• Calculate exponents next.
• Multiplication and division have equal priority and are performed from left to right.
• Addition and subtraction also have equal priority and are performed from left to right.
• Complicated calculations should be completed one step at a time.
• When evaluating algebraic expressions, substitute known values for variables before completing the calculation.
• Parentheses are especially useful when substituting negative numbers.
• A fraction bar acts as a grouping symbol.
• Calculators follow the order of operations, but expressions must be entered correctly.
• The order of operations is essential in algebra, geometry, science, engineering, finance, and everyday calculations.

Did You Know?

The order of operations is a mathematical convention. It allows people around the world to interpret the same expression consistently. Without such conventions, even a short expression such as 2 + 3 × 4 could be ambiguous—and mathematics would need many more brackets to make every calculation clear.