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.