Order of Operations with Integers
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What you will learn
- 1Apply PEMDAS to evaluate integer expressions with parentheses, exponents, and four operations
- 2Identify the correct next step in a partially evaluated expression
- 3Explain why multiplication and division share a level and work left to right
- 4Avoid the common error of adding before multiplying
Mini Lesson
When an expression mixes parentheses, exponents, and the four basic operations, mathematicians need to agree on the order so everyone gets the same answer. The rule is called the order of operations, and most learners remember it as PEMDAS (or GEMDAS, where G stands for grouping symbols).
PEMDAS / GEMDAS — One Step at a Time
- P — Parentheses (or any grouping symbol such as brackets) come first.
- E — Exponents next: 3^2 means 3 x 3, and 2^3 means 2 x 2 x 2.
- M and D — Multiplication and Division share a level. Work them left to right in the order they appear.
- A and S — Addition and Subtraction share the bottom level. Work them left to right too.
- Do ONE operation per step. Rewrite the whole expression after each step.
Common Error: Adding Before Multiplying
In 3 + 4 x 2, multiplication beats addition. The correct path is 3 + 4 x 2 = 3 + 8 = 11. Someone who adds first would get 7 x 2 = 14, which is wrong. The order of operations is what stops two people from getting two different answers.
Worked Example: 3 + 4 x 2^2
- Exponent first: 2^2 = 4. Expression becomes 3 + 4 x 4.
- Multiplication next: 4 x 4 = 16. Expression becomes 3 + 16.
- Addition last: 3 + 16 = 19.
- Final answer: 19.
Working with Integers
- Negative numbers follow the same order rules — handle the sign as you go.
- (-3)^2 means (-3) x (-3) = 9 (parentheses make the whole -3 the base).
- Without the parentheses, -3^2 means the opposite of 3^2, which is -9.
- When a multiplication and a division sit next to each other, work LEFT to RIGHT — not multiplication first.
Words to know
order of operations
The agreed order for evaluating an expression: parentheses, exponents, multiply and divide left to right, add and subtract left to right.
parentheses
Grouping symbols ( ) that tell you to do whatever is inside first.
exponent
A small raised number that says how many copies of the base are multiplied, like the 2 in 5^2.
evaluate
To find the value of an expression by performing all of its operations in the correct order.
expression
A combination of numbers, operations, and grouping symbols that has a value but no equals sign.
Tip: do ONE operation per line and rewrite the rest of the expression. This prevents you from forgetting a number and makes mistakes easy to spot.
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