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Video: Order of Operations: PEMDAS

Tim Burnham

We work through a series of examples in order to explain how to apply the rules of the order of operations to consistently evaluate arithmetic expressions such as 11 − 10 / 5 ∗ 4 + 9. We use the acronym PEMDAS to help remember the order.

15:36

Video Transcript

In this video, we’re gonna be looking at the order of operations and we’re gonna be using a system called PEMDAS. You may have heard of BODMAS or BIDMAS, but we’re using PEMDAS.

When you’ve got complicated expressions to evaluate, PEMDAS will help you to decide which bits to do first, which bits to do second, and so on. So we’re gonna go through the rules, and then we’re gonna look at some examples. The first on the list is P, for parentheses. And really it’s about parentheses or grouping

So some examples of those would be when you have the round brackets, the round parentheses, like three plus five. You might have square brackets, four minus two. Or you might have a fraction and the numerator implies some grouping; you have to keep the three and six together, so that’s implied as a grouping. So for example, we’ve got an expression like this five plus three is in parentheses, then times six. So five plus three we have to evaluate first, and five plus three is obviously eight. And when we evaluated the parentheses, obviously we can get rid of them. So that then turned into eight times six, which is forty-eight.

So another example is five plus three all over two. Well this means five plus three divided by two. But because the five plus three is the numerator, this implies that we have to do this first; it’s grouped together. So it is almost as though we’ve got parentheses around the numerator. So doing that first, we end up with eight over two. Obviously, eight over two, we don’t need the parenthesis anymore. And that eight over two really means eight divided by two, and eight divided by two is four. One more example three plus two in parentheses times seven minus two in parentheses. So P is top of the list, so we’ve got to evaluate the parentheses first. But we’ve got two, so what do we do? Well we do the left one first, and then we do work our way right. So we do the left one first, then we do the right one. And three plus two is five, so that’s the same as five times seven minus two. Now we can evaluate the right-hand parentheses. And seven minus two is five, so we’ve got five times five, which is twenty-five.

So it’s important to remember with PEMDAS that if two different terms have got equal billing, they’ve both got parentheses around them, then you work from left to right. So we did this one first, and then we did this one. Now E, exponents. For example, 𝑥 squared, 𝑥 to the power of two. And after you’ve evaluated any parentheses, then you need to evaluate your exponents, and be careful to make sure you apply the exponents only to their bases. Let’s see some examples. two plus three squared. Well that’s got parentheses in it, so we need to evaluate those first. And two plus three is equal to five, so that’s equal to five squared, which means five times five, which gives us an answer of twenty-five.

But what if we’d have started off with just two plus three squared without any parentheses? Well, no parentheses, nothing to do there. So next we’re on to our exponents. So we have to evaluate three squared which is three times three, which is nine, and two plus nine is eleven. So it’s important to see that that exponent only applies to this base, the three, not to the whole of the two plus three. We’d have needed parentheses to force that to happen. Okay, what about this one, negative three squared? Well, there are no parentheses, so we don’t have to worry about that. Next on the list is exponents, so we’ve got to do the three squared. What of our negative sign in front of the three there? That could either be interpreted as negative one times three squared or it could be interpreted as take away three squared, just negative three squared. In either case, it’s either a multiplication or subtraction; they’re lower down the list than exponents. So first of all we have to do just the three squared. three squared is nine, so we’ve got the negative of nine. So there’s our answer: negative nine. Now some of you may have been thinking, “Hold on! negative three squared is negative three times negative three. That should have been positive nine!” But the way we wrote it, we didn’t write negative three all squared we just wrote the negative of three squared. Now if we’d have written negative three in parentheses, then it’s the whole of those parentheses that are squared. So that means negative three times negative three, and that is nine. So you really do have to pay attention to what you’re doing with these.

Next on the list is multiplication or division. Again it’s really important to remember, these have got the same level of precedence multiplication or division; they’re both equally prioritised in this list. But you just work from left to right throughout the expression that you’re trying to evaluate. So for example, fifteen divided by five times two. Because the M comes before the D in PEMDAS doesn’t mean to say that you have to do multiplication first always. You work from left to right, remember? They’ve got equal priority, so first we’re going to do the fifteen divided by five, which is three, so that’s three times two. And now we’ve only got one thing to do, three times two, which is six. So let’s have a look at this one: two times three plus four times five. Well there are no parentheses, there are no exponents, but we’ve got two lots of multiplication, so we’re gonna work from left to right. First of all, we’re gonna do this one. And two times three is six, so that translates to six plus four times five. And then we’re gonna do this one, and four times five is twenty, so we’ve got six plus twenty, which is twenty-six.

And last on the list is addition or subtraction. Again, they both have equal precedence in this list, so just work from left to right through the expression. So for example, one plus two minus three. We’ve got an addition and we’ve got a subtraction. So we’re just gonna work our way from left to right through that expression. one plus two is three, so we’ve got three minus three. And three minus three is zero, which is our answer. Let’s look at another one then: five minus two plus three. We’ve got a subtraction and we’ve got an addition. Just because it says AS in PEMDAS doesn’t mean to say you have to do the addition first; we have to work from left to right through that expression. So first of all, we’re gonna do the five take away two, which is three. So we’ve got three plus three. Now we can just add three and three to give us six. Now we said if you’ve got addition and subtraction, they’re both equally prioritised, so you work from left to right. But by putting parentheses around the right-hand two terms in the expression, we can force people to do those first. So these parentheses means that we’re going to look inside here; we’re gonna do this two plus three before we then move on to do the five take away. Well two plus three is five, so five take away five, which is zero.

Let’s look at a few general examples now then. eleven take away ten divided by five times four plus three squared. Well we don’t have any parentheses, so we don’t need to do anything up there. Exponents? Yep, we’ve got some exponents, so we better do those first. And three squared is equal to nine, so we’ve now got eleven take away ten divided by five times four plus nine. So have we got any multiplication or division? Yep, we have, so we better do that next. We’ve got one division and one multiplication. And we have to work from left to right, so I’m gonna do the division first, and then we’re gonna do the multiplication. So ten divided by five is two, so we’ve got eleven minus two times four plus nine. Now we’re gonna do the multiplication two times four. So we’ve got eleven minus two times four, so that’s eleven minus eight, and we’ve got plus nine on the end. So we’ve done the P and the E and the M and the D, so now we’re on to the addition and subtraction. And again, we’re gonna go from left to right. So first, we’re gonna do this one, and then we’re gonna do this one. And eleven minus eight is three, so we’ve got three plus nine, which is equal to twelve.

Let’s see another example then four times open parenthesis two times two close parenthesis divided by four times open parenthesis two plus two close parenthesis So we’ve got two parentheses; I’m gonna evaluate out the left one first and then the right one. And two times two is four, so that leaves us with four times four divided by four times two plus two. So evaluating that second parenthesis now gives us four times four divided by four times four. So we’ve done the parentheses. There aren’t any exponents, so we’re now moving on to multiplication and division. Now they’ve got equal precedence, we just have to work from left to right. So first we’re gonna do this one, then we’re gonna do this one, and then we’re gonna do this one. So we get an answer of sixteen.

Now it’s worth noting that there’s a common mistake that people make. Lots of people notice you got parentheses here, so in fact they say, “Oh let’s evaluate the whole of that bit,” and then they see these parentheses here and evaluate the whole of that bit. So they say two times two is four times four is sixteen and two plus two is four times four is sixteen . So they do sixteen divided by sixteen, and they get the answer one. That is wrong! So this working from left to right is an important rule. Now if we had wanted the answer to be equal to one, what we’d need to have done is put a parenthesis round the whole of that first expression and a parenthesis around the whole of that expression there. So that would’ve forced us to do the parentheses within the parentheses first, and then the left-hand one, and then the parentheses within this parentheses first, and then that one. That would’ve forced us to get sixteen divided by sixteen is equal to one.

Right, let’s look at this one then. We’ve got five minus two plus three all squared minus three minus six squared divided by two. So we’ve gotta evaluate the parentheses first. And looking inside there, we’ve only got addition and subtraction so we’re gonna work our way from left to right within that first parenthesis. So first of all, we’ve got five minus two, which is three, so that becomes three plus three. Now we’ve got three plus three is equal to six, so we’ve got six squared minus three minus six squared divided by two. So that’s dealt with our parentheses. Next, we look for exponents. And we’ve got two: this one and this one. So evaluating the left one first, that becomes thirty-six. And now evaluating the second one, that is also six squared. So that’s also thirty-six. Well no more exponents, so we’re now moving on to multiplication and division. Well, there’s no multiplication but there is some division going on here: thirty-six divided by two is eighteen. So that just leaves us with some addition and subtraction to do, in fact only subtraction. So we’ve got thirty-six take away three take away eight, so we do the first one first. thirty-six take away three is thirty-three, and thirty-three take away eighteen is fifteen, which is our answer.

Now look at this example, we’ve got two sets of parentheses: an inner one and an outer one. So we’re gonna evaluate the inner one first. And within that, we’ve got twenty-five divided by five take away two, so we’ve got division and we’ve got subtraction. So looking down PEMDAS, we have to do the division before the subtraction. So it looks like the first thing that we’ve done was evaluate a division and that’s way down the list below parentheses. But remember, we are working on the inner parenthesis, so twenty-five divided by five is five. So we’ve now gotta evaluate five minus two; they’re in a parenthesis again. And five minus two is three, so that in our parenthesis here turns out to be three squared. So now we’ve still got a parenthesis, so we’ve got to evaluate that. So it’s the whole of three squared minus one all squared. So I’ve got to evaluate the parenthesis first, three squared minus one. So within that, we’ve got to do the three squared first because that’s an exponent, and then we’re gonna be subtracting one because subtraction is lower down the list than exponents. So that’s three squared is nine, and nine minus one is eight. So that paren- set of parentheses has evaluated itself to be eight, so we’re now left with eight squared, which is sixty-four. So if you’ve got multiple parentheses, if you got parentheses inside parentheses, you have to evaluate those first. And because inside parentheses you could still have some quite complicated expressions in themselves, the actual first operation you carry out in a calculation might be multiplication, division, addition, or subtraction even. You have to evaluate the contents of the parentheses before you go on to evaluate other parts of the expression

Right, let’s look at one last example then. So this is pretty complicated. So in terms of parentheses, we’ve got two sets of parentheses, so we can evaluate the left-hand one first. And ten minus five gives us five. Now actually, there is an implied pair of parentheses around here; this line here on a fraction that divides numerator from the denominator is implying a grouping on everything in the numerator. So we can sort of say that they’re implied parentheses there. So I’m actually gonna evaluate those next. So I’ve got five squared plus five; gonna do the squared first, and then I’m gonna add the five. And twenty-five plus five is thirty. So that’s dealt with the parentheses there. I can now evaluate this expression here so negative two all squared means negative two times negative two, which is positive four. So I’ve dealt with the parentheses, I can now look for exponents, and there is one over here: two squared. two squared is four So that’s all of our parentheses and exponents dealt with. Now it’s multiplication and division. Well I’ve got an implied division here: thirty divided by six, which is five. So now I’ve got only addition and subtraction left. I’m gonna work from the left, so I’m gonna do five plus four first and then I’m gonna subtract four. So five plus four is nine, and nine take away four is five. Well we’ve looked a quite a few examples there. Hopefully that’s giving you enough flavour of PEMDAS to enable you to evaluate any expression that you see.