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## Order of operations

## Video transcript

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.