Order of Operations (PEMDAS)

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Apply the correct order of operations to evaluate expressions.
  • Distinguish between subtraction and the negative sign.
  • Use grouping symbols correctly (parentheses, brackets, fraction bars).
  • Interpret expressions with exponents, negatives, and nested grouping.

Key Ideas

The order of operations is remembered by PEMDAS:

  1. P – Parentheses / grouping symbols
  2. E – Exponents
  3. M – Multiplication
  4. D – Division
  5. A – Addition
  6. S – Subtraction

Multiplication and division have equal priority (work left to right).
Addition and subtraction also have equal priority (work left to right).

PEMDAS hierarchy diagram showing the order of operations.
Important

A fraction bar acts like a grouping symbol. For

\[ \frac{a + b}{c + d} \]

evaluate the numerator and denominator separately before dividing.

Common Problem Types

1. Evaluating Expressions

Example:
\[ 3 + 6 \times 4 \]

Step-by-step:

  1. Multiply: \(6 \times 4 = 24\)
  2. Add: \(3 + 24 = 27\)

2. Expressions with Parentheses

Example:
\[ (3 + 6) \times 4 \]

Step-by-step:

  1. Parentheses: \(3 + 6 = 9\)
  2. Multiply: \(9 \times 4 = 36\)

3. Negative Sign vs Subtraction

Example:
\[ -3^2 \quad \text{vs} \quad (-3)^2 \]

Step-by-step:

  • \(-3^2\): exponent applies to 3 only → \(3^2 = 9\) → final answer: \(-9\)
  • \((-3)^2\): exponent applies to the entire number → \((-3)(-3) = 9\)

These are not the same.

4. Exponents Before Multiplication

Example:
\[ 2 \cdot 3^2 \]

Step-by-step:

  1. Exponent: \(3^2 = 9\)
  2. Multiply: \(2 \cdot 9 = 18\)

5. Multi-Level Grouping

Example:
\[ [2(3 + 4)]^2 \]

Step-by-step:

  1. Parentheses: \(3 + 4 = 7\)
  2. Multiply: \(2 \cdot 7 = 14\)
  3. Exponent: \(14^2 = 196\)

Fraction Bar as a Grouping Symbol

Example:
Evaluate
\[ \frac{6 + 2}{4 - 1} \]

Step-by-step:

  1. Numerator: \(6 + 2 = 8\)
  2. Denominator: \(4 - 1 = 3\)
  3. Fraction: \(\frac{8}{3}\)

Strategies

  • Evaluate numerator and denominator separately when a fraction bar is present.
  • Keep negative signs attached to numbers unless parentheses explicitly change grouping.
  • Follow left-to-right for both multiplication/division and addition/subtraction.
  • Rewrite complex expressions one line at a time to maintain clarity.

Worked Examples

Example 1

Question:
\[ 8 - 3 \times 2^2 \]

Step-by-step Solution:

  1. Exponent: \(2^2 = 4\)
  2. Multiply: \(3 \times 4 = 12\)
  3. Subtract: \(8 - 12 = -4\)

Answer: \(-4\)

Example 2

Question:
\[ \frac{12 - 4}{2 + 1} \]

Step-by-step Solution:

  1. Numerator: \(12 - 4 = 8\)
  2. Denominator: \(2 + 1 = 3\)
  3. Fraction: \(\frac{8}{3}\)

Answer: \(\frac{8}{3}\)

Example 3

Question:
\[ 5 - (-3)^2 \]

Step-by-step Solution:

  1. Exponent: \((-3)^2 = 9\)
  2. Substitute: \(5 - 9\)
  3. Subtract: \(5 - 9 = -4\)

Answer: \(-4\)

WarningCommon Mistakes
  • Doing addition before multiplication due to misreading PEMDAS.
  • Ignoring the left-to-right rule for multiplication and division.
  • Treating \(-3^2\) as \(9\) instead of \(-9\).
  • Forgetting that a fraction bar groups numerator and denominator.
  • Misreading expressions with negatives and exponents due to missing parentheses.

Practice Problems

Evaluate each expression using correct order of operations.

  1. \(7 + 2 \cdot 5\)
  2. \((7 + 2) \cdot 5\)
  3. \(-4^2\)
  4. \((-4)^2\)
  5. \(\dfrac{10 - 3}{1 + 2}\)
  6. \(3 + 6 \div 2\)
  7. \(3(2 + 5)^2\)

1. \(7 + 2 \cdot 5\)
Step 1: Multiply → \(2 \cdot 5 = 10\)
Step 2: Add → \(7 + 10 = 17\)
Answer: \(17\)

2. \((7 + 2) \cdot 5\)
Step 1: Parentheses → \(7 + 2 = 9\)
Step 2: Multiply → \(9 \cdot 5 = 45\)
Answer: \(45\)

3. \(-4^2\)
Step 1: Exponent applies to 4 → \(4^2 = 16\)
Step 2: Apply negative sign → \(-16\)
Answer: \(-16\)

4. \((-4)^2\)
Step 1: Parentheses group the negative → \((-4)(-4)\)
Step 2: Multiply → \(16\)
Answer: \(16\)

5. \(\dfrac{10 - 3}{1 + 2}\)
Step 1: Numerator → \(7\)
Step 2: Denominator → \(3\)
Answer: \(\dfrac{7}{3}\)

6. \(3 + 6 \div 2\)
Step 1: Division → \(6 \div 2 = 3\)
Step 2: Add → \(3 + 3 = 6\)
Answer: \(6\)

7. \(3(2 + 5)^2\)
Step 1: Parentheses → \(2 + 5 = 7\)
Step 2: Exponent → \(7^2 = 49\)
Step 3: Multiply → \(3 \cdot 49 = 147\)
Answer: \(147\)

Summary

  • PEMDAS organizes operations, but multiplication/division and addition/subtraction occur left to right.
  • Fraction bars and parentheses act as grouping symbols.
  • Parentheses determine how exponents apply to negative numbers.
  • Rewriting expressions step-by-step reduces errors.
  • Treat a fraction bar like parentheses—simplify top and bottom separately.
  • MD and AS follow the left-to-right rule.
  • Use parentheses to clarify expressions such as \((-3)^2\).
  • For nested grouping, simplify the innermost expression first.