Order of Operations Study Guide
Introduction to Order of Operations
The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a specific degree.
—ARISTOTLE (384–322 B.C.)
Please excuse my dear aunt Sally, and you'll breeze through this lesson about the order of operations. Think PEMDAS is overrated? Think again!
When you get a messy mathematical expression that involves every operation under the sun, you must remember to perform the operations in the correct order. This order, often referred to as PEMDAS, is:
Parentheses → Exponents → Multiplication and Division (from left to right) → Addition and Subtraction (from left to right)
First, perform any math operations located inside parentheses.
Tip:Often, in expressions, there are grouping symbols—usually shown as parentheses—which are used to make a mathematical statement clear.
Then, calculate any exponents.
Tip:An exponent is a number that tells you how many times a number is multiplied by itself: 23 = 2 × 2 × 2. For more on exponents.
Next, solve the multiplication and division from left to right. Finally, complete any addition or subtraction from left to right.
PEMDAS is often remembered with the phrase Please Excuse My Dear Aunt Sally. You may want to create your own personal sentence to remember the order of operations.
You may be wondering why you really need to follow the order of operations. Does it really matter? Let's look at what happens when you ignore PEMDAS and attack a problem in order of appearance:
- 9 + 8 × 2 – 3 × 2
- 17 × 2 – 3 × 2
- 34 – 3 × 2
- 31 × 2
Is this answer correct? Because you ignored PEMDAS, this is not the right answer. No worries—now proceed in the correct order. There are no parentheses or exponents, so we need to do any multiplication or division first from left to right.
- 9 + 8 × 2 – 3 × 2
- 9 + 16 – 3 × 2
- 9 + 16 – 6
Now, complete the addition and subtraction from left to right.
- 9 + 16 – 6
- 25 – 6
Wow—without using the order of operations, the answer wasn't even close to the actual value! Remember, take your time and carry out each operation in the correct order.
Let's look at another example.
22 + (6 – 5) – (3 + 3) × 3 =
Begin by completing any work inside the parentheses.
(6 – 5) = 1 and (3 + 3) = 6
The original problem is now as follows:
22 + 1 – 6 × 3 =
Calculate the exponent: 22 = 4. Now, you can further simplify the problem.
4 + 1 – 6 × 3 =
The next stage of PEMDAS, the MD, indicates that you should do all the multiplication and division from left to right. There is no division to worry about, but there is multiplication: 6 × 3. This equals 18. So, your problem is now:
4 + 1 – 18
You're almost there! Perform all addition and subtraction. Remember to do this from left to right.
4 + 1 = 5 and 5 – 18 = –13
So, 22+ (6 – 5) – (3 + 3) × 3 = –13.
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