Hey, welcome. My name is Anthony Varela. And today, I'm going to introduce the order of operations. So we're going to do two main things. First, we're going to talk about what the order of operations is. And secondly, we're going to use the order of operations to evaluate some
Expressions.
So first, we need to know what the order of operations is. And PEMDAS is a widely-used acronym to help us remember what the order of operations is. So PEMDAS-- an acronym to remember the order of operations.
And some folks out there even like to use this as a sentence, "Please Excuse My Dear Aunt Sally," to remember the letters that make up PEMDAS. But remember, PEMDAS helps us remember operations. So we're going to write this out.
P is for parentheses. E is for exponents. M is for multiplication. D is for division. A is for addition. And S is for subtraction. But there are a couple of notes that I'd like to make about each of these.
When we're talking about parentheses, this also includes other grouping symbols, like braces and brackets. And there are even some other operations that imply some parentheses, but we're not going to get into detail here. So we're going to write this down, though, that the P for parentheses also includes other grouping symbols. Please be aware of that.
Now, with exponents, exponents can also be called "powers," but we're also including radicals or roots in this step as well. So that's going to be important to remember, that the E for exponents also includes radicals. Now, with multiplication and division, we perform these as we read them left to right, not always multiplication before division. So that's going to be important to remember, that the M and the D for multiplication and division are evaluated as we see them reading left to right.
And that same idea applies with addition and subtraction. We perform these as we see them, reading left to right, not always addition before subtraction in our order. So the A and the S for addition and subtraction are evaluated reading them left to right. So now that we know what the order of operations is, let's put this into practice and evaluate some expressions.
So here we have 32 divided by 4 times 2 plus 7. So taking a look at the operations that we have here, we have division, we have multiplication, and we have addition. And looking at our order of operations, remembering PEMDAS, I'm tempted to say, OK, we need to evaluate the multiplication first, because it comes first in our order of multiplication. But remember that multiplication and division are evaluated as we read them left to right, so we're actually going to evaluate 32 divided by 4 first.
So 32 divided by 4 equals 8. So now I'm just rewriting my expression, but I've put an 8 for the 32 divided by 4. So now we have 8 times 2 plus 7.
Now, according to our order of operations, then I evaluate the multiplication next, before the addition. So 8 times 2 is 16. So I'm going to write in 16, then plus 7. Now we just have one operation left-- the addition. So this is evaluated, then, to 23, following the proper order of operations.
Now let's see how this works differently. We notice we have the same numbers. We have the same operations. The numbers are put in the same order, but I have these parentheses here. And we know that parentheses comes first in our order of operations.
So here, we actually are going to evaluate the multiplication first, not because it's before division when we're comparing multiplication and division reading left to right. It's because we're doing the parentheses first. So we know that 4 times 2 is 8, so we can rewrite this as 32 divided by 8 plus 7. So now I'm going to evaluate that division before I add.
So 32 divided by 8 equals 4. And then we can add 7 on to that and we get 11. So notice that these two gave us very different answers here, even though we're using the same numbers-- we're using the same operations, but we performed these in a different order, because they had these parentheses, here which changed the order of everything.
Let's go through another example here. We have 17 minus 4 plus 3 times 2. So taking a look at our order of operations, what are we going to do first? Well, we're going to do that multiplication first-- this 3 times 2 needs to be evaluated before we move on to anything else. So I can rewrite this as 17 minus 4 plus 6.
So now what am I going to do? Am I going to do these subtraction first or the addition first? Well, taking a look at PEMDAS, I'm tempted to say, let's add the 4 and the 6 before we subtract 4 from 17. But remember, with addition and subtraction, we evaluate these as we see them left to right.
So I actually see the subtraction first. So I'm going to take away 4 from 17 first. So I have 13 plus 6. And that sums up to equal 19. So my entire expression is evaluated to 19.
Let's see how this is different. We are again using the same numbers, using the same operations, but I've put in some parentheses here. How are we going to handle this? Well, what's inside the parentheses? There's still quite a bit.
We see addition and multiplication. We have to evaluate all of this first before we even deal with this 17. So 4 plus 3 times 2. I'm going to evaluate the multiplication first. It comes before addition in the order of operations. So I can rewrite this as 17 minus 4 plus 6, 6 being 3 times 2.
So now I'm going to add 4 and 6, and I can write this as 17 minus 10. And lastly, I get to compute the subtraction here. 17 minus 10 equals 7. So once again, you can see how just adding in some parentheses into your original expression definitely changes how you perform the order of operations, and it affects what number you get at the very end.
So let's review our notes. Today, we talked about the order of operations and we used this acronym, PEMDAS. It's an acronym to remember the order of operations-- parentheses, exponents, multiplication and division, addition and subtraction. And there are a couple of notes about each of these.
The P for parentheses also includes other grouping symbols. The E for exponents also includes radicals. And the M and D for multiplication and division is performed as we see it left to right.
And same with addition and subtraction. We perform these as we see them left to right. So thanks for watching this video on an introduction to the order of operations. Hope to see you next time.