Order of Operations & PEMDAS Cheat Sheet

A printable reference covering parentheses, exponents, multiplication, division, addition, subtraction, grouping symbols, and left-to-right rules for grades 5-7.

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Order of operations tells you the correct order to simplify a math expression. Students need this cheat sheet because the same numbers can give different answers if the steps are done in the wrong order. PEMDAS helps organize each step so expressions are solved clearly and consistently. This reference is useful for arithmetic, fractions, decimals, variables, and word problems. The main rule is to simplify grouping symbols first, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right. Parentheses, brackets, braces, and fraction bars can all act as grouping symbols. Multiplication does not always come before division, and addition does not always come before subtraction. When two operations have the same priority, work from left to right.

Key Facts

PEMDAS means Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
Simplify grouping symbols first, so $(18 - 6) \div 3 = 12 \div 3 = 4$ .
Evaluate exponents before multiplication, division, addition, or subtraction, so $2 + 3^2 = 2 + 9 = 11$ .
Multiplication and division have equal priority, so solve them from left to right: $24 \div 6 \times 2 = 4 \times 2 = 8$ .
Addition and subtraction have equal priority, so solve them from left to right: $15 - 6 + 2 = 9 + 2 = 11$ .
A fraction bar groups the entire numerator and denominator, so $\frac{8 + 4}{3} = \frac{12}{3} = 4$ .
For nested grouping symbols, simplify the innermost group first, such as $2[3 + (8 - 5)] = 2[3 + 3] = 12$ .
Check each step by rewriting the whole expression until only one number remains.

Vocabulary

Expression: A mathematical phrase with numbers, variables, and operations but no equal sign.
Operation: A math action such as addition, subtraction, multiplication, division, or using an exponent.
Grouping Symbols: Symbols such as parentheses, brackets, braces, and fraction bars that show which part of an expression to simplify first.
Exponent: A small raised number that tells how many times to use the base as a factor, such as $4^3$ .
Product: The result of multiplying two or more factors.
Quotient: The result of dividing one number by another number.

Common Mistakes to Avoid

Multiplying before simplifying parentheses: this is wrong because all grouping symbols must be completed before outside operations, as in $4(10 - 7) = 4 \times 3$ .
Always doing multiplication before division: this is wrong because multiplication and division have the same priority, so $18 \div 3 \times 2$ is solved from left to right.
Always doing addition before subtraction: this is wrong because addition and subtraction have the same priority, so $20 - 5 + 1$ is solved from left to right.
Forgetting that a fraction bar is a grouping symbol: this is wrong because $\frac{6 + 4}{2}$ means simplify $6 + 4$ before dividing by $2$ .
Applying an exponent to the wrong number: this is wrong because $-3^2$ means $-(3^2)$ , while $(-3)^2$ means $(-3)(-3)$ .

Practice Questions

1 Simplify $8 + 3 \times 4 - 6 \div 2$ .
2 Simplify $5^2 - 2(6 + 3)$ .
3 Simplify $\frac{12 + 8}{4} + 3^2$ .
4 Explain why $24 \div 4 \times 2$ should be solved from left to right instead of multiplying first.