Video Transcript: Order of Operations

Intro to order of operations 

In this video we're going to talk a little bit about order of operations. And I want you to pay  close attention because really everything else that you're going to do in mathematics is going  to be based on you having a solid grounding in order of operations. So what do we even mean  when we say order of operations? So let me give you an example. The whole point is so that we  have one way to interpret a mathematical statement. So let's say I have the mathematical  statement 7 plus 3 times 5. Now if we didn't all agree on order of operations, there would be  two ways of interpreting this statement. You could just read it left to right, so you could say  well, let me just take 7 plus 3, you could say 7 plus 3 and then multiply that times 5. And 7 plus 3  is 10, and then you multiply that by 5. 10 times 5, it would get you 50. So that's one way you  would interpret it if we didn't agree on an order of operations. Maybe it's a natural way. You just go left to right. Another way you could interpret it you say, I like to do multiplication before I do addition. So you might interpret it as -- I'll try to color code it -- 7 plus -- and you do the 3 times  5 first. 7 plus 3 times 5, which would be 7 plus 3 times 5 is 15, and 7 plus 15 is 22. So notice, we  interpreted this statement in two different ways. This was just straight left to right doing  addition then the multiplication. This way we did the multiplication first then the addition, we  got two different answers, and that's just not cool in mathematics. If this was part of some  effort to send something to the moon because two people interpreted it a different way or  another one computer interpreted one way and another computer interpreted it another way,  the satellite might go to mars. So this is just completely unacceptable, and that's why we have  to have an agreed upon order of operations. An agreed upon way to interpret this statement.  So the agreed upon order of operations is to do parentheses first -- let me write it over here --  then do exponents. If you don't know what exponents are don't worry about it right now. In this video we're not going to have any exponents in our examples, so you don't really have to worry  about them for this video. Then you do multiplication -- I'll just right mult, short for  multiplication -- then you do multiplication and division next, they kind of have the same level  of priority. And then finally you do addition and subtraction. So what does this order of  operations -- let me label it -- this right here, that is the agreed upon order of operations. If we  follow these order of operations we should always get to the same answer for a given  statement. So what does this tell us? What is the best way to interpret this up here? Well we  have no parentheses -- parentheses look like that. Those little curly things around numbers. We  don't have any parentheses here. I'll do some examples that do have parentheses. We don't  have any exponents here. But we do have some multiplication and division or we actually just  have some multiplication. So we'll order of operations, do the multiplication and division first.  So it says do the multiplication first. That's a multiplication. So it says do this operation first. It  gets priority over addition or subtraction. So if we do this first we get the 3 times 5, which is 15,  and then we add the 7. The addition or subtraction -- I'll do it here, addition, we just have  addition. Just like that. So we do the multiplication first, get 15, then add the 7, 22. So based  upon the agreed order of operations, this right here is the correct answer. The correct way to  interpret this statement. Let's do another example. I think it'll make things a little bit more  clear, and I'll do the example in pink. So let's say I have 7 plus 3 -- I'll put some parentheses there -- times 4 divided by 2 minus 5 times 6. So there's all sorts of crazy things here, but if you just  follow the order of operations you'll simplify it in a very clean way and hopefully we'll all get the

same answer. So let's just follow the order of operations. The first thing we have to do is look  for parentheses. Are there parentheses here? Yes, there are. There's parentheses around the 7  plus 3. So it says let's do that first. So 7 plus 3 is 10. So this we can simplify, just looking at this  order operations, to 10 times all of that. Let me copy and paste that so I don't have to keep re writing it. So that simplifies to 10 times all of that. We did our parentheses first. Then what do  

we do? There are no more parentheses in this expression. Then we should do exponents. I don't  see any exponents here, and if you're curious what exponents look like, an exponent would  look like 7 squared. You'd see these little small numbers up in the top right. We don't have any  exponents here so we don't have to worry about it. Then it says to do multiplication and  division next. So where do we see multiplication? We have a multiplication, a division, a  multiplication again. Now, when you have multiple operations at the same level, when our  order of operations, multiplication and division are the same level, then you do left to right. So  in this situation you're going to multiply by 4 and then divide by 2. You won't multiply by 4  divided by 2. Then we'll do the 5 times 6 before we do the subtraction right here. So let's figure  out what this is. So we'll do this multiplication first. We could simultaneously do this  multiplication because it's not going to change things. But I'll do things one step at a time. So  the next step we're going to do is this 10 times 4. 10 times 4 is 40. 10 times 4 is 40, then you  have 40 divided by 2 and it simplifies to that right there. Remember, multiplication and  division, they're at the exact same level so we're going to do it left to right. You could also  express this as multiplying by 1/2 and then it wouldn't matter the order. But for simplicity,  multiplication and division go left to right. So then you have 40 divided by 2 minus 5 times 6.  So, division, you just have one division here, you want to do that. You have this division and you have this multiplication, they're not together so you can actually kind of do them  simultaneously. And to make it clear that you do this before you do the subtraction because  multiplication and division take priority over addition and subtraction, we could put  parentheses around them to say look, we're going to do that and that first before I do that  subtraction, because multiplication and division have priority. So 40 divided by 2 is 20. We're  going to have that minus sign, minus 5 times 6 is 30. 20 minus 30 is equal to negative 10. And  that is the correct interpretation of that. So I want to make something very, very, very clear. If  you have things at the same level, so if you have 1 plus 2 minus 3 plus 4 minus 1. So addition  and subtraction are all the same level in order of operations, you should go left to right. So you  should interpret this as 1 plus 2 is 3, so this is the same thing as 3 minus 3 plus 4 minus 1. Then  you do 3 minus 3 is 0 plus 4 minus 1. Or this is the same thing as 4 minus 1, which is the same  thing as 3. You just go left to right. Same thing if you have multiplication and division, they're at the same level. So if you have 4 times 2 divided by 3 times 2, you do 4 times 2 is 8 divided by 3  times 2. And you say 8 divided by 3 is, well, we got a fraction there. It would be 8/3. So this  would be 8/3 times 2. And then 8/3 times to is equal to 16 over 3. That's how you interpret it. You don't do this multiplication first or divide the 2 by that and all of that. Now the one time where  you can be loosey-goosey with order of operations, if you have all addition or all multiplication.  So if you have 1 plus 5 plus 7 plus 3 plus 2, it does not matter what order you do it in. You can do  the 2 plus 3, you can go from the right to the left, you can go from the left to the right, you  could start some place in between. If it's only all addition. And the same thing is true if you have all multiplication. It's 1 times 5 times 7 times 3 times 2. It does not matter what order you're 

doing it. But it's only with all multiplication or all addition. If there was some division in here, if  there's some subtraction in here, you're best off just going left to right. 

Order of operations examples: exponents 

So I have six different expressions here, and what I want you to do is pause this video and try to  calculate the value of each of these expressions. I'm assuming you've given a go at it. Now let's  work through them. So when we see something like this, we have to remember our order of  operations. We have 2 times 3 squared, and we have to remember that the first thing we would  need to think about are the parentheses. I'll just write paren for short. Then we worry about  exponents. Then we will worry about multiplication and division, and actually let me write it  this way. We worry about multiplication and division. And then we worry about addition and  subtraction. So in this expression right over here, there are no parentheses, so we do the  exponents first. So we calculate what 3 squared is. 3 times 3 is 9, so this becomes 2 times 9,  which is equal to 18. Now let's look at this one, and this one is interesting, because they have--  it looks like the same expression, but now there are parentheses. And because of these  parentheses, we're going to do the multiplication before we take the exponent. So 2 times 3 is  going to be 6, and we're going to take that to the second power. So that's 6 times 6, which is  equal to 36. Now let's think about this one right over here. Once again, we want to do our  multiplication and our division first. So we have a division right over here. 81/9 is the same  thing as 81 divided by 9, and that's going to be 9. And then we have-- so it becomes 1 plus 5  times 9. Now we want to do the multiplication before we do the addition, so we're going to do  our 5 times 9, which is 45. So this becomes 1 plus 45, which of course is equal to 46. Now let's  tackle this one right over here. So, we would want to do the exponents first. So, 1 squared, well  that's just going-- let me do this in a different color. 1 squared is just going to be equal to 1, so  that's just going to be equal to 1. And so you have 2 times 4 plus 1. What should you do? Should  you add first or do the multiplication first? Well multiplication takes precedence over addition,  so you're going to do the 2 times 4 first. 2 times 4 is 8, so you're going to have 8 plus 1, which of  course is equal to 9. Now you have a very similar expression, but you have parentheses. So  that's going to force you to do what's in the parentheses before you take the exponent. But  within the parentheses we have multiplication and addition, and we have to remember that we  do the multiplication first. So we're going to do the 2 times 4 first, so that's going to be 8 plus 1  to the second power. 8 plus 1 is 9, so that's 9 to the second power. 9 squared is the same thing  as 9 times 9, which is equal to 81. Now we have one more right over here that looks very similar to this one, except, once again, we have parentheses that's making us do the addition first.  Without parentheses, we would do the multiplication and the division first. But here, we see  that 1 plus 5 is 6, and then we have this 81/9, which is 9. So this simplifies to 6 times 9, which of  course is equal to 54. 

Order of operations example 

We're asked to simplify 8 plus 5 times 4 minus, and then in parentheses, 6 plus 10 divided by 2  plus 44. Whenever you see some type of crazy expression like this where you have parentheses  and addition and subtraction and division, you always want to keep the order of operations in  mind. Let me write them down over here. So when you're doing order of operations, or really  when you're evaluating any expression, you should have this in the front of your brain that the 

top priority goes to parentheses. And those are these little brackets over here, or however you  want to call them. Those are the parentheses right there. That gets top priority. Then after that, you want to worry about exponents. There are no exponents in this expression, but I'll just write 

it down just for future reference: exponents. One way I like to think about it is parentheses  always takes top priority, but then after that, we go in descending order, or I guess we should  say in-- well, yeah, in descending order of how fast that computation is. When I say fast, how  fast it grows. When I take something to an exponent, when I'm taking something to a power, it  grows really fast. Then it grows a little bit slower or shrinks a little bit slower if I multiply or  divide, so that comes next: multiply or divide. Multiplication and division comes next, and then  last of all comes addition and subtraction. So these are kind of the slowest operations. This is a  little bit faster. This is the fastest operation. And then the parentheses, just no matter what,  always take priority. So let's apply it over here. Let me rewrite this whole expression. So it's 8  plus 5 times 4 minus, in parentheses, 6 plus 10 divided by 2 plus 44. So we're going to want to  do the parentheses first. We have parentheses there and there. Now this parentheses is pretty  straightforward. Well, inside the parentheses is already evaluated, so we could really just view  this as 5 times 4. So let's just evaluate that right from the get go. So this is going to result in 8  plus-- and really, when you're evaluating the parentheses, if your evaluate this parentheses,  you literally just get 5, and you evaluate that parentheses, you literally just get 4, and then  they're next to each other, so you multiply them. So 5 times 4 is 20 minus-- let me stay  consistent with the colors. Now let me write the next parenthesis right there, and then inside of it, we'd evaluate this first. Let me close the parenthesis right there. And then we have plus 44.  So what is this thing right here evaluate to, this thing inside the parentheses? Well, you might  be tempted to say, well, let me just go left to right. 6 plus 10 is 16 and then divide by 2 and you  would get 8. But remember: order of operations. Division takes priority over addition, so you  actually want to do the division first, and we could actually write it here like this. You could  imagine putting some more parentheses. Let me do it in that same purple. You could imagine  putting some more parentheses right here to really emphasize the fact that you're going to do  the division first. So 10 divided by 2 is 5, so this will result in 6, plus 10 divided by 2, is 5. 6 plus 5.  Well, we still have to evaluate this parentheses, so this results-- what's 6 plus 5? Well, that's 11.  So we're left with the 20-- let me write it all down again. We're left with 8 plus 20 minus 6 plus 5, which is 11, plus 44. And now that we have everything at this level of operations, we can just go  left to right. So 8 plus 20 is 28, so you can view this as 28 minus 11 plus 44. 28 minus 11-- 28  minus 10 would be 18, so this is going to be 17. It's going to be 17 plus 44. And then 17 plus 44--  I'll scroll down a little bit. 7 plus 44 would be 51, so this is going to be 61. So this is going to be  equal to 61. And we're done! 

Worked example: Order of operations (PEMDAS) 

Now that we've got the basics of order of operations out of the way, let's try to tackle a really  hairy and beastly problem. So here, we have all sorts of parentheses and numbers flying  around. But in any of these order of operations problems, you really just have to take a deep  breath and remember, we're going to do parentheses first. Parentheses. P for parentheses.  Then exponents. Don't worry if you don't know what exponents are, because this has no  exponents in them. Then you're going to do multiplication and division. They're at the same 

level. Then you do addition and subtraction. So some people remember PEMDAS. But if you  remember PEMDAS, remember multiplication, division, same level. Addition and subtraction,  also at the same level. So let's figure what the order of operations say that this should evaluate  to. So the first thing we're going to do is our parentheses. And we have a lot of parentheses  here. We have this expression in parentheses right there, and then even within that we have  these parentheses. So our order of operations say, look, do your parentheses first, but in order  to evaluate this outer parentheses-- this orange thing-- we're going to have to evaluate this  thing in yellow right there. So let's evaluate this whole thing. So how can we simplify it? Well, if  we look at just inside of it, the first thing we want to do is simplify the parentheses inside the  parentheses. So you see this 5 minus 2 right there? We're going to do that first no matter what.  And that's easy to evaluate. 5 minus 2 is 3. And so this simplifies to-- I'll do it step by step. Once  you get the hang of it, you can do multiple steps at once. So this is going to be 7 plus 3 times  the 5 minus 2, which is 3. And all of those have parentheses around it. And of course, you have  all the stuff on either side-- the divide 4-- no. Oops. That's not what I want. I wanted to copy  and paste. I want to copy and paste that right there. So copy, then-- no, that's giving me the  wrong thing. It would've been easier-- let me just rewrite it. That's the easiest thing. I'm having  technical difficulties. So divided by 4 times 2. And on this side, you had that 7 times 2 plus this  thing in orange parentheses there. Now, at any step you just look again. We always want to do  parentheses first. Well, you keep wanting to do and is there really no parentheses left? So we  have to evaluate this parentheses in orange here. So we have to evaluate this thing first. But in  order to evaluate this thing, we have to look inside of it. And when you look inside of it, you  have 7 plus 3 times 3. So if you just had 7 plus 3 times 3, how would you evaluate it? Well, look  back to your order of operations. We're inside the parentheses here, so inside of it there are no  longer any parentheses. So the next thing we should do is-- there are no exponents. There is  multiplication. So we do that before we do any addition or subtraction. So we want to do the 3  times 3 before we add the 7. So this is going to be 7 plus-- and the 3 times 3 we want to do first.  We want to do the multiplication first. 7 plus 9. That's going to be in the orange parentheses.  And then you have the 7 times 2 plus that, on the left hand side. You have the divided by 4  times 2 on the right hand side. And now this-- the thing in parentheses-- because we still want  to do the parentheses first. Pretty easy to evaluate. What's 7 plus 9? 7 plus 9 is 16. And so  everything we have simplifies to 7 times 2 plus 16 divided by 4 times 2. Now we don't have any  parentheses left, so we don't have to worry about the P in PEMDAS. We have no E, no  exponents in this. So then we go straight to multiplication and division. We have a  multiplication-- we have some multiplication going on there. We have some division going on  here, and a multiplication there. So we should do these next, before we do this addition right  there. So we could do this multiplication. We could do that multiplication. 7 times 2 is 14. We're  going to wait to do that addition. And then here we have a 16 divided by 4 times 2. That gets  priority of the addition, so we're going to do that before we do the addition. But how do we  evaluate that? Do we do the division first, or the multiplication first? And remember, I told you  in the last video, when you have 2-- when you have multiple operations of the same level-- in  this case, division and multiplication-- they're at the same level. You're safest going left to right. Or you should go left to right. So you do 16 divided by 4 is 4. So this thing right here-- simplify  16 divided by 4 times 2. It simplifies to 4 times 2. That's this thing in green right there. And then 

we're going to want to do the multiplication next. So this is going to simplify to-- because  multiplication takes priority over addition-- this simplifies to 8. And so you get 14-- this 14 right  here-- plus 8. And what's 14 plus 8? That is 22. That is equal to 22. And we are done. 

Order of operations example 

Simplify negative 1 times this expression in brackets, negative 7 plus 2 times 3 plus 2 minus 5, in parentheses, squared. So this is an order of operations problem. And remember, order of  operations, you always want to do parentheses first. Parentheses first. Then you do exponents.  Exponents. And there is an exponent in this problem right over here. Then you want to do  multiplication. Multiplication and division. And then finally, you do addition and subtraction. So let's just try to tackle this as best we can. So first, let's do the parentheses. We have a 3 plus 2  here in parentheses, so we can evaluate that to be equal to 5. And let's see, we could do other  things in other parts of this expression that won't affect what's going on right here in the  parentheses. We have this negative 5 squared. Or I should just, we're subtracting a 5 squared.  We want to do the exponent before we worry about it being subtracted. So this 5 squared over  here we can rewrite as 25. And so let's not do too many steps at once. So this whole thing will  simplify to negative 1. And then in brackets, we have negative 7 plus 2 times 5. And then, 2  times 5. And then close brackets. Minus 25. Now, this thing-- we want to do multiplication. You  could say, hey, wait. I still have a parentheses here. Why don't I do that first? But when you just  evaluate what's inside of this parentheses, you just get a negative 7. It doesn't really change  anything. So we can just leave this here as a negative 7. And this expression. We do want to  evaluate this whole expression before we do anything else. I mean, we could distribute this  negative 1 and all of that, but let's just do straight up order of operations here. So let's evaluate  this expression. We want to do multiplication before we add anything. So we get 2 times 5 right over there. 2 times 5 is 10. That is 10. So our whole expression becomes-- and normally, you  wouldn't have to rewrite the expression this many times. But we're going to do it this time just  to make sure no one gets confused. So it becomes negative 1 times negative 7 plus 10. Plus 10.  And we close our brackets. Minus 25. Now, we can evaluate this pretty easily. Negative 7 plus  10. We're starting at negative 7. So I was going to draw a number line there. So we're starting--  let me draw a number line. So we're starting at negative 7. So the length of this line is negative  7. And then, we're adding 10 to it. We're adding 10 to it. So we're going to move 10 to the right.  If we move 7 to the right, we get back to 0. And then we're going to go another 3 after that. So  we're going to go 7, 8, 9, 10. So that gets us to positive 3. Another way to think about it is we  are adding integers of different signs. We can view the sum as going to be the difference of the  integers. And since the larger integer is positive, our answer will be positive. So you could  literally just view this as 10 minus 7. 10 minus 7 is 3. So this becomes a 3. And so our entire  expression becomes negative 1. Negative 1 times. And just to be clear, brackets and  parentheses are really the same thing. Sometimes people will write brackets around a lot of  parentheses just to make it a little bit easier to read. But they're really just the same thing as  parentheses. So these brackets out here, I could just literally write them like that. And then I  have a minus 25 out over here. Now, once again, you want to do multiplication or division  before we do addition and subtraction. So let's multiply the negative 1 times 3 is negative 3. 

And now we need to subtract our 25. So negative 3 minus 25. We are adding two integers of the same sign. We're already at negative 3 and we're going to become 25 more negative than that.  So you can view this as we're moving 25 more in the negative direction. Or you could view it as  3 plus 25 is 28. But we're doing it in the negative direction, so it's negative 28. So this is equal to  negative 28. And we are done.