Order of Operations

Applied Math · 3 min read

Order of Operations

The Standard Sequence

Mathematical expressions can contain multiple operations. The order of operations establishes which calculations to perform first. Without this standard sequence, the same expression could yield different answers.

The sequence is commonly remembered as PEMDAS:

Parentheses (and other grouping symbols)
Exponents
Multiplication and Division (left to right)
Addition and Subtraction (left to right)

Multiplication and division have equal priority; perform them in order from left to right. Addition and subtraction also have equal priority; perform them left to right.

Parentheses and Grouping Symbols

Parentheses indicate which operations to perform first. Whatever is inside parentheses gets calculated before applying outside operations.

Worked Example

3 × (4 + 5) = 3 × 9 = 27. Without parentheses: 3 × 4 + 5 = 12 + 5 = 17. The parentheses changed the answer significantly.

Brackets [ ] and braces { } serve the same purpose as parentheses. When nested, work from innermost to outermost.

Worked Example

2 × [3 + (4 × 2)] = 2 × [3 + 8] = 2 × 11 = 22.

Fraction bars act as grouping symbols too. Everything above the bar is one group; everything below is another. Calculate each part before dividing.

Worked Example

(6 + 4) / (3 - 1) = 10 / 2 = 5.

Exponents

Exponents indicate repeated multiplication. The base is the number being multiplied; the exponent tells how many times.

Worked Example

4³ = 4 × 4 × 4 = 64 (4 is the base, 3 is the exponent) 2⁵ = 2 × 2 × 2 × 2 × 2 = 32

Calculate exponents after parentheses but before multiplication, division, addition, or subtraction.

Worked Example

3 + 2² × 4 = 3 + 4 × 4 = 3 + 16 = 19.

Special cases to remember:

Any number to the power of 1 equals itself: 7¹ = 7

Any number to the power of 0 equals 1: 5⁰ = 1

Negative bases: (-2)² = 4 (the negative is squared) but -2² = -4 (only 2 is squared, then negated)

Working Through Complex Expressions

Apply the order of operations systematically.

Worked Example

20 - 3 × (4 + 2)² ÷ 6

Step 1: Parentheses first. 4 + 2 = 6. Expression becomes 20 - 3 × 6² ÷ 6. Step 2: Exponents next. 6² = 36. Expression becomes 20 - 3 × 36 ÷ 6. Step 3: Multiplication and division, left to right. 3 × 36 = 108. Expression becomes 20 - 108 ÷ 6. Then 108 ÷ 6 = 18. Expression becomes 20 - 18. Step 4: Subtraction. 20 - 18 = 2.

Final answer: 2.

Workplace Applications

Order of operations appears in formulas for calculating pay, costs, measurements, and many other workplace quantities.

Worked Example

Calculating overtime pay Formula: Regular pay + (Overtime hours × Regular rate × 1.5)

If regular pay is $400, overtime hours are 8, and regular rate is $10: $400 + (8 × $10 × 1.5) = $400 + ($120) = $520.

The parentheses ensure you calculate all the overtime components before adding to regular pay.

Worked Example

Material cost calculation Total cost = Base cost + (Quantity × Unit price) - Discount

If base cost is $50, quantity is 20, unit price is $3, and discount is $15: $50 + (20 × $3) - $15 = $50 + $60 - $15 = $95.

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Order of Operations

The Standard Sequence

The sequence is commonly remembered as PEMDAS:

Parentheses (and other grouping symbols)
Exponents
Multiplication and Division (left to right)
Addition and Subtraction (left to right)

Multiplication and division have equal priority; perform them in order from left to right. Addition and subtraction also have equal priority; perform them left to right.

Parentheses and Grouping Symbols

Parentheses indicate which operations to perform first. Whatever is inside parentheses gets calculated before applying outside operations.

Worked Example

3 × (4 + 5) = 3 × 9 = 27. Without parentheses: 3 × 4 + 5 = 12 + 5 = 17. The parentheses changed the answer significantly.

Brackets [ ] and braces { } serve the same purpose as parentheses. When nested, work from innermost to outermost.

Worked Example

2 × [3 + (4 × 2)] = 2 × [3 + 8] = 2 × 11 = 22.

Fraction bars act as grouping symbols too. Everything above the bar is one group; everything below is another. Calculate each part before dividing.

Worked Example

(6 + 4) / (3 - 1) = 10 / 2 = 5.

Exponents

Exponents indicate repeated multiplication. The base is the number being multiplied; the exponent tells how many times.

Worked Example

4³ = 4 × 4 × 4 = 64 (4 is the base, 3 is the exponent) 2⁵ = 2 × 2 × 2 × 2 × 2 = 32

Calculate exponents after parentheses but before multiplication, division, addition, or subtraction.

Worked Example

3 + 2² × 4 = 3 + 4 × 4 = 3 + 16 = 19.

Special cases to remember:

Any number to the power of 1 equals itself: 7¹ = 7

Any number to the power of 0 equals 1: 5⁰ = 1

Negative bases: (-2)² = 4 (the negative is squared) but -2² = -4 (only 2 is squared, then negated)

Working Through Complex Expressions

Apply the order of operations systematically.

Worked Example

20 - 3 × (4 + 2)² ÷ 6

Final answer: 2.

Workplace Applications

Order of operations appears in formulas for calculating pay, costs, measurements, and many other workplace quantities.

Worked Example

Calculating overtime pay Formula: Regular pay + (Overtime hours × Regular rate × 1.5)

If regular pay is $400, overtime hours are 8, and regular rate is $10: $400 + (8 × $10 × 1.5) = $400 + ($120) = $520.

The parentheses ensure you calculate all the overtime components before adding to regular pay.

Worked Example

Material cost calculation Total cost = Base cost + (Quantity × Unit price) - Discount

If base cost is $50, quantity is 20, unit price is $3, and discount is $15: $50 + (20 × $3) - $15 = $50 + $60 - $15 = $95.

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