What exactly is the order of operations?
The order of operations is really just the simple order or steps to solve an equation. In other words, it would be the "special" order for solving the very confusing equations with a bunch of parenthesis and numbers. Here's an example:
18-{12+8³[(3+2•3•5)6]14÷2}3²=?
Can you solve this equation?
You may think this is easy. But is it really? Let's try it:
18-{12+8³[(3+2•3•5)6]14÷2}3²=?
↓
6+512•3+2•3•5•6•7•27=26,467,56
No,no,no! That is very, very wrong! This problem sure is a toughie! But it's alright. We'll come back to it later...when we're experts! Right now, we have to focus on learning the order of operations first.
So before we begin, we must learn the order of operations.
The most important thing to know about the order of operations is the order? Get it? You see, the "order" of operations? Oh, never mind...Anyway, here's the order:
-
Parenthesis
-
Exponents
-
Multiply & Divide (from left to right)
-
Add & Subtract (from left to right)
If you're thinking, "Wow, that's a lot to remember," you can always use PEMDAS.
PEMDAS is just a little phrase to make it easier to remember the order of operations. What does it stand for, you ask? Let me show you:
Please
Excuse
My
Dear
Aunt
Sally
Of course it doesn't have to be that phrase. You can make up your own if it helps. But it still has to start with the same letters as the one I gave you there. Because if you look closely enough, you can see that PEMDAS also spells out the order
of operations. Take another look:
Parenthesis
Exponents
Multiply
Divide
Add
Subtract
Ahh, finally! Now that we know the order of operations, let's begin!
But we're not quite ready to start attacking that first tricky equation. So let's work on an easier one for now. How about this one:
13-5•(5-2)+(16÷2)•4²=?
Remember to use PEMDAS! First comes the P for parenthesis ( )!
First, we do whatever is in the parenthesis. Now then, inside the parenthesis, we see 5 minus 2 and 16 divided by 2. We'll solve those first:
13-5•(5-2)+(16÷2)•4²=?
↓
13-5•3+8•4²=?
Although not present in this particular equation, but in some cases, such as the tricky equation at the top of this page, there will be some brackets [ ] and maybe even some braces { }.
Please note that although they are different symbols, they all belong to the same family or category. It's just that if we used parenthesis for all three of those, well, that would be pretty confusing. The brackets and the braces usually are found (if they can be found) around the parenthesis. In those cases, you would still do the parenthesis first. Then, after you've done the parenthesis, you would work your way out to the brackets. And then finally you would solve the numbers in the braces. Like this (it's not the equation we're working on, but let's jump to this one really quickly):
{3+5[8+2-(9-3)+4]-12}+2=?
First, we solve what's in the parenthesis, remember? (well, you should...I just mentioned it a few seconds ago...) So let's do it:
{3+5+[8+2-(9-3)+4]-12}+2=?
↓
{3+5+[8+2-6+4]-12}+2=?
Okay, now that we've done those, we have to move on out to the brackets. See those numbers in there:
{3+5+[8+2-6+4]-12}+2=?
↓
{3+5+[4+4]-12}+2=?
↓
{3+5+8-12}+2=?
Now that those first brackets and parenthesis are finished, time to move to the final steps of the parenthesis rule. Do you see that we're slowly working our way out of every parenthesis? If you don't, watch the second to last step! This is where we do the braces:
{3+5+8-12}+2=?
↓
{16-12}+2=?
↓
4+2=?
See? Now that wasn't hard, was it? And look, now the only thing left to do is figure out whatever is outside of the all those parenthesis! So the real last step is this:
4+2=?
↓
?=6
Now, back to where we left off before the whole parenthesis thing...Ah, yes! The next step after parenthesis, is E for exponents!
Always after the parenthesis, is the exponents. You know, the little numbers on top of other numbers that look like this: y³ Well, that's our next step in the order of operations. Let's continue on the problem we stopped at. Find the exponent in our previous equation an solve it:
13-5•3+8•4²=?
↓
13-5•3+8•4•4=?
↓
13-5•3+8•16=?
With the exponents taken care of, the next step in PEMDAS is M and D. What does that stand for? Multiplication and division!
Now, these two steps don't have a specific order with each other, although in PEMDAS, multiplication comes before division. That doesn't affect the order though. For these two, we always work with whatever comes first, from left to right. So if division comes before, we'll do that first. But if it's multipication, we'll do that. No sweat! So it's kind of like "First come, first serve." Ready? Here, multiplication comes first (either way, there is no division anymore, so...):
13-5•3+8•16=?
↓