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).

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.