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( ) × ÷ + PEMDAS

Order of
Operations

Master the rules for evaluating expressions with parentheses, exponents, multiplication, division, addition, and subtraction.

SSAT Upper Level: Quantitative

What Is the Order of Operations?

The order of operations is a set of rules that tells you which calculations to perform first when evaluating a mathematical expression. Without these rules, the same expression could give different answers.

Why Does It Matter?

Universal Agreement
Every mathematician, scientist, and computer evaluates the same expression the same way.
Prevents Ambiguity
Without a standard order, \( 2 + 3 \times 5 \) could mean \(25\) or \(17\): one fixed answer eliminates confusion.
Test Essential
SSAT Quantitative questions frequently test whether you apply operations in the correct sequence.

The PEMDAS Framework

P Parentheses E Exponents M / D Multiply / Divide A / S Add / Subtract Highest priority ← → Lowest priority Left to Right M and D share the same level. Work left to right, not M before D. Left to Right A and S share the same level. Work left to right, not A before S. Remember: "Please Excuse My Dear Aunt Sally" P → E → MD (left to right) → AS (left to right)

The Rule Stated Formally

1
Parentheses / Brackets: Evaluate everything inside grouping symbols first: \( ( \;\; ), \; [ \;\; ], \; \{ \;\; \} \).
2
Exponents: Evaluate powers and roots: \( x^n, \; \sqrt{x} \).
3
Multiplication and Division: Work left to right. Neither takes priority over the other.
4
Addition and Subtraction: Work left to right. Neither takes priority over the other.

Same-Level vs Different-Level

Same Level Different Levels
Example \( \times \) and \( \div \) \( \times \) and \( + \)
Rule Left to right Higher level first
Common Error Doing \(\times\) before \(\div\) Doing \(+\) before \(\times\)
Correct Approach \(12 \div 3 \times 2 = 8\) \(3 + 4 \times 2 = 11\)

How to Apply PEMDAS

1
Scan the expression for grouping symbols (parentheses, brackets). Simplify innermost groups first.
2
Evaluate exponents from left to right.
3
Perform multiplication and division as they appear left to right.
4
Perform addition and subtraction as they appear left to right.
5
Double-check by re-reading the expression. Confirm no step was skipped.

Example 1: Basic

Evaluate: \( \;8 + 2 \times 5 - 3\)

Step 1: Multiply first
\(8 + {\color{#fbbf24}2 \times 5} - 3 \rightarrow 8 + {\color{#fbbf24}10} - 3\)
Step 2: Add (left to right)
\({\color{#4ade80}8 + 10} - 3 \rightarrow {\color{#4ade80}18} - 3\)
Step 3: Subtract
\({\color{#818cf8}18 - 3} \rightarrow {\color{#818cf8}15}\)
Answer: 15

Example 2: Parentheses and Exponents

Evaluate: \(\;(3 + 5)^2 \div 4 - 1\)

Step 1: Parentheses
\({\color{#818cf8}(3 + 5)}^2 \div 4 - 1 \rightarrow {\color{#818cf8}8}^2 \div 4 - 1\)
Step 2: Exponents
\({\color{#a78bfa}8^2 = 64} \rightarrow 64 \div 4 - 1\)
Step 3: Divide
\({\color{#fbbf24}64 \div 4 = 16} \rightarrow 16 - 1\)
Step 4: Subtract
\({\color{#4ade80}16 - 1 = 15}\)

Example 3: Nested Grouping

Evaluate: \(\;2 \times [3 + (4^2 - 10)] \div 2\)

1
Innermost parentheses: \(4^2 - 10\). Exponent first: \(16 - 10 = \) \(6\)
2
Brackets: \(3 + 6 = \) \(9\)
3
Multiply and Divide left to right: \(2 \times 9 = 18\), then \(18 \div 2 = \) \(9\)

Example 4: The Fraction Bar

Evaluate: \(\;\dfrac{6 + 2 \times 4}{20 - 3^2}\)

1
Numerator: Multiply first: \(2 \times 4 = 8\). Then add: \(6 + 8 = \) \(14\)
2
Denominator: Exponent first: \(3^2 = 9\). Then subtract: \(20 - 9 = \) \(11\)
3
Divide: \(\frac{14}{11} = \) \(\frac{14}{11}\)

Special Grouping Symbols

Absolute Value \(| \;\; |\)
Treat \(|x|\) like parentheses: simplify inside first, then take the absolute value. Example: \(|3 - 7| = |-4| = 4\)
Radical \(\sqrt{\phantom{x}}\)
The bar of a radical groups everything underneath. Example: \(\sqrt{9 + 16} = \sqrt{25} = 5\), not \(3 + 4\).
Fraction Bar
Treat numerator and denominator as separate groups: simplify each before dividing.

Exponent Traps

Expression Common Wrong Answer Correct Answer
\(-3^2\) \(9\) \(-9\)
\((-3)^2\) \(-9\) \(9\)
\(2 \times 3^2\) \(36\) \(18\)

Common Mistakes

✕ Left-to-Right Only
Ignoring PEMDAS and evaluating strictly left to right: \(3 + 4 \times 2 \neq 14\).
✕ M Before D Always
Multiplying before dividing regardless of position: \(12 \div 3 \times 2 \neq 2\). The correct answer is \(8\).
✕ Negative Exponents
Squaring the negative sign: \(-5^2 = -25\), not \(25\). Parentheses change everything.

Summary Cheat Sheet

Priority Operation Rule
1st Parentheses / Brackets Innermost first
2nd Exponents / Roots Left to right
3rd Multiplication and Division Left to right (same level)
4th Addition and Subtraction Left to right (same level)

Question 1 of 5

What is the value of \(\;5 + 3 \times 2^2\;\)?

Question 2 of 5

Evaluate: \(\;(6 + 2) \times 3 - 4 \div 2\;\)

Question 3 of 5

What is the value of \(\;-4^2 + (-4)^2\;\)?

Question 4 of 5

Evaluate: \(\;\dfrac{10 + 2 \times 3}{8 - 2^2}\;\)

Question 5 of 5

Simplify: \(\;3 \times [2 + (5^2 - 20)] - 7\;\)