Order of Operations at a Glance

The order in which we complete a mathematical equation or expression is super important. Doing the steps of a problem out of order can lead to a totally different answer, even if we do the calculations correctly. So, for everyone to get the same answer, the order of operations was developed. We'll be using these almost every day of our mathematical careers from now on.

All equations and expressions must be completed in the following order. There's even a handy acronym to help us remember it.

PEMDAS

Parentheses: simplify everything inside the parentheses.
Exponents: simplify any exponents (see Powers and Roots.)
Multiplication & Division: multiply and divide from left to right.
Addition & Subtraction: add and subtract from left to right.

PEMDAS will be our friend for life. If we have trouble remembering our new BFF's name, we can try out this memory trick:

"Please Excuse My Dear Aunt Sally"

Look Out: even though M comes before D, multiplication and division are equals; we'll need to complete these operations from left to right. The same is true for addition and subtraction. Think of them as equal partners.

How to Do It

Let's try simplifying 4(3 + 7)² – 25(100 ÷ 5) ÷ 50. Where do we start?

According to our new buddy PEMDAS, we simplify all the stuff inside the parentheses (P) first.

4(3 + 7)² – 25(100 ÷ 5) ÷ 50 =
4(10)² – 25(20) ÷ 50

Nice. Next up is the E in PEMDAS, which stands for exponents. There's an exponent of 2 on that 10, so our next step is to find (10)².

4(10)² – 25(20) ÷ 50 =
4(100) – 25(20) ÷ 50

So far, so good. Our next letters are M and D, so now we rock multiplication and division. Remember, these two letters are equal partners, so we do all our multiplying and dividing from left to right.

4(100) – 25(20) ÷ 50 =
400 – 25(20) ÷ 50 =
400 – 500 ÷ 50 =
400 – 10

Our very last letters in PEMDAS are A (addition) and S (subtraction), which we also do from left to right. There isn't any addition here, so we finish up by subtracting 10 from 400.

400 – 10 = 390

We're done!

Examples
Exercises

Order of Operations Example 1

What is 5 – 3 + 6 ÷ 2?
Since there are no parentheses or exponents, division is our first step (the D in PEMDAS). 5 – 3 + 6 ÷ 2 = 5 – 3 + 3
Next, simplify the first addition/subtraction operation. 5 – 3 + 3 = 2 + 3
Finally, simplify the last addition/subtraction operation. 2 + 3 = 5 All done! Now try to do the problem in your head from left to right. The answer would then be 4, which is obviously not correct. See? Order matters!

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Order of Operations Example 2

What is 5(6 + 9 ÷ 3 – 1)² ?
First, simplify the parentheses. Since there are many operations in the parentheses, follow the order of operations inside them, too. In this problem, division comes first. 5(6 + 9 ÷ 3 – 1)² = 5(6 + 3 – 1)²
Next, simplify the addition and subtraction from left to right. 5(6 + 3 – 1)² = 5(9 – 1)² = 5(8)²
Now, tackle the exponents. 5(8)² = 5(64)
Finally, multiply. 5(64) = 320 Awesome. We're done!

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Order of Operations Example 3

Simplify .
In this problem, treat the numerator and denominator like they're in separate parentheses, and the fraction bar as a division sign. You could rewrite this problem as: . Now simplify each set of parentheses using the correct order of operations. =
=
Finally, divide. = Yahoo!

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Order of Operations Example 4

Simplify [3(6 – 4)]².
Here, we have two sets of parentheses: the regular ones and the brackets. Simplify the innermost set first. [3(6 – 4)]² = [3(2)]²
Next, simplify the stuff inside the brackets. [3(2)]² = [6]²
Finally, simplify the exponents. [6]² = 36 Finished!