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Lesson 5: More on order of operations

# Order of operations review

The order of operations are a set of rules for how to evaluate expressions. They make sure everyone gets to the same answer. Many people memorize the order of operations as PEMDAS (parentheses, exponents, multiplication/division, and addition/subtraction).
The order of operations are one set of agreements for how to evaluate expressions. They make sure everyone gets to the same value.
start color #543b78, start text, G, end text, end color #543b78rouping: We evaluate what's inside grouping symbols first, before anything else. For example, 2, times, start color #543b78, left parenthesis, 3, plus, 1, right parenthesis, end color #543b78, equals, 2, times, 4, equals, 8.
Two common types of grouping symbols are parentheses and the fraction bar.
start color #0c7f99, start text, E, end text, end color #0c7f99xponents: We evaluate exponents before multiplying, dividing, adding, or subtracting. For example, 2, times, start color #0c7f99, 3, squared, end color #0c7f99, equals, 2, times, 9, equals, 18.
start color #9e034e, start text, M, end text, end color #9e034eultiplication and start color #9e034e, start text, D, end text, end color #9e034eivision: We multiply and divide before we add or subtract. For example, 1, plus, start color #9e034e, 4, divided by, 2, end color #9e034e, equals, 1, plus, 2, equals, 3.
start color #a75a05, start text, A, end text, end color #a75a05ddition and start color #a75a05, start text, S, end text, end color #a75a05ubtraction: Lastly, we add and subtract.
Many people memorize the order of operations as start color #7854ab, start text, G, end text, end color #7854ab, start color #0c7f99, start text, E, end text, end color #0c7f99, left parenthesis, start color #9e034e, start text, M, D, end text, end color #9e034e, right parenthesis, left parenthesis, start color #a75a05, start text, A, S, end text, end color #a75a05, right parenthesis (pronounced as it's spelled), where the "G" is for grouping, the "E" is for exponents, and so on.
Important note: When we have more than one of the same type of operation, we work from left to right. This can matter when subtraction or division are on the left side of your expression, like 4, minus, 2, plus, 3 or 4, divided by, 2, times, 3 (see example 3 below to understand why this matters).

## Example 1

Evaluate 6, times, 4, plus, 2, times, 3.
There are no parentheses or exponents, so we jump straight to multiplication and division.
empty space, 6, times, 4, plus, 2, times, 3${}$
equals, start color #9e034e, 6, times, 4, end color #9e034e, plus, 2, times, 3Multiply start color #9e034e, 6, end color #9e034e and start color #9e034e, 4, end color #9e034e.
equals, 24, plus, start color #9e034e, 2, times, 3, end color #9e034eMultiply start color #9e034e, 2, end color #9e034e and start color #9e034e, 3, end color #9e034e.
equals, start color #a75a05, 24, plus, 6, end color #a75a05Add start color #a75a05, 24, end color #a75a05 and start color #a75a05, 6, end color #a75a05.
equals, 30... and we're done!
Notice: We took care of all multiplication before doing the addition. If we had done 24, plus, 2 before multiplying 2, times, 3, we would have gotten the wrong answer.

## Example 2

Evaluate 6, squared, minus, 2, left parenthesis, 5, plus, 1, plus, 3, right parenthesis.
empty space, 6, squared, minus, 2, left parenthesis, 5, plus, 1, plus, 3, right parenthesis${}$
equals, 6, squared, minus, 2, left parenthesis, start color #543b78, 5, plus, 1, plus, 3, end color #543b78, right parenthesisAdd start color #543b78, 5, plus, 1, plus, 3, end color #543b78 inside the parentheses first.
equals, start color #0c7f99, 6, end color #0c7f99, start superscript, start color #0c7f99, 2, end color #0c7f99, end superscript, minus, 2, left parenthesis, 9, right parenthesisFind start color #0c7f99, 6, squared, end color #0c7f99, which is 6, dot, 6, equals, 36.
equals, 36, minus, start color #9e034e, 2, left parenthesis, 9, right parenthesis, end color #9e034eMultiply start color #9e034e, 2, end color #9e034e and start color #9e034e, 9, end color #9e034e.
equals, start color #a75a05, 36, minus, 18, end color #a75a05Subtract 18 from 36.
equals, 18... and we're done!

## Example 3

Evaluate 7, minus, 2, plus, 3.
One correct way to do this is to work from left to right.
CorrectIncorrect
\begin{aligned}&7-2+3\\\\=&5+3\\\\=&8\end{aligned}\begin{aligned}&7-2+3\\\\=&7-5\\\\=&2\end{aligned}
Remember: Even though "A" comes before "S" in GE(MD)(AS), that doesn't mean we need to add before we subtract. Addition and subtraction are at the same "level" in the order of operations. The same is true of multiplication and division.