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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.

Practice

Problem 1
• Current
2, plus, 12, divided by, 2, times, 3, equals

Want to practice more problems like these? Check out this introductory exercise and these more challenging exercises: exercise one and exercise two.

Want to join the conversation?

• the last question is absolutely wrong, the answer should be 12 because
-10 + 8 = -2
-2^2 = -4
8 - -4 = 12
• In the second step, there should be a parenthesis around the -2 (as it is the sum of -10 and +8 from the previous step). Because the negative sign is also included, -2 rather than +2 is squared. This results in +4. Here are the steps:
8 − (−10 + 8)^2
=8 - (-2)^2
=8 - 4
=4
Edit: fixed typos and added explanation.
• Is there another way to say PEMDAS?( it could be another word) I really don't like the sound of it...
• It's normally "Please excuse my dear Aunt Sally", even though I don't know what the aunt did. :D Hope this helped!
• A way to remember PEMDAS is "Please Excuse My Dear Aunt Sally"
Go Aunt Sally and all her arithmatic glory!
• ACTUALLY ITS SPEMDAS. slice pizzas except my dads awesome slice. the s is for substitution in algibra
• I don't understand when there is division then straight up multiplacation
• look like 20/5=? but we know our multiplication so we could say 4x5=20 so the missing number is 4 and 20/5=4
• what if its 25^10x12(50+12)
• I think you have to use a calculator.
First calculate (50+12). And calculate 25^10 with your calculator. Then we only have multiplication symbols. So, just calculate from left to right with your calculator.
• What is the best way to solve a quotation with a exponent when the exponent is 3 or above? For example if the exponent of the number 2 is 3, should I do 2 times 2 which is 4 and then do 4 times 2 which equals 8 again? How am I suppose to do that with bigger exponents?
• Remember exponents are just a simpler way of multiplication where instead of saying 2*2*2, we can say 2^3, or in word form, two to the third power, so to answer question yes you could multiply 2*2=4, and then multiply 4*2=8.
(1 vote)
• apparently it's now PEDMAS my teacher told me that, I'm not quite sure though. I'm just a little confused weather its PEDMAS or PEMDAS but I guess it wouldn't really matter
• No its Please excuse my dear aunt sally
• i am canadian I do bedmas
• So you probably can get the same answers as us then
• 87747873483-8374574+95848498658386484=
• 9.5848586e+16
(1 vote)
• : Deal with what's in the parentheses first. While working inside the parentheses, follow PEMDAS as well. So -(2^2+6)... first, I'm going to assume the "+6" is not part of the exponent. It's hard to tell with the way it is typed.
There are no more parentheses so inside the one set, we'll move to "E". 2^2 becomes 4. Now our expression looks like:
-(4+6)
There are no multiplication or division inside the parentheses so we'll move on the AS: 4+6 = 10 so now the expression looks like:
-(10…