In this video we're going to talk about something called the order of operations.
Now, an operation is basically something that takes two values in, and it will combine them
in some way, and then spit out a result, an output.
So for instance, addition is an operation.
If you took three plus two, it would combine them, and it would give you five, and that's
a new answer.
But there's something we forget a lot of times about operations, and that's the fact that
they can only accept two numbers at a time.
And so things get a little confusing if you have more than one term in your expression.
For instance, if you had three times four, minus two this could mean two different answers
depending on what order you apply these operations.
If you did the multiplication first, and did three times four, and then subtracted two
you'd get a different answer than if you did four minus two, and then multiplied by three.
So one way we can eliminate this confusion is by using ah, parenthesis very heavily.
Uhm, if we applied parenthesis, parenthesis would group two of the numbers together to
be done first, and then we would apply the other operation afterwards.
So for instance, if you said parenthesis, three times four, and then subtract two, this
would be very clear to somebody reading your work that you obviously mean twelve minus
two, which would give you ten as an answer.
But if you meant three times parenthesis, four minus two, then this would mean three
times six, cause four minus two gives you ah, I'm sorry, three times two, rather, four
minus two is two, and three times two would give you six.
And, which is a different answer, and so you say, well that clears it up then, I guess
we'll just use parenthesis all over the place.
But see that, that's not very practical though, it's a good idea but when you start getting
larger expressions like this, that have many, many operations, then if you tried to go with
the parenthesis�s only way of doing things, you would have literally seven or eight different
sets of parenthesis telling you what order to do these operations which can only accept
two inputs at a time.
And so this is what the order of operations is going to help with, it's going to help
us reduce how many parenthesis we have all over the place.
Because it's going to tell us the order, the, the usual order, or the default order in which
operations are typically completed.
Okay, so here's, here's the order in which things are done.
Uhm, if you see parenthesis, parenthesis takes precedence over everything.
So if you see parenthesis, that's done regardless, first of all.
But then after that, we'll have exponents that are done first.
Then multiplication and division, and then addition and subtraction.
And the reason I have these four highlighted is actually I have a separate little comment
to make about these.
Just because multiplication is listed before division, and addition is listed before subtraction,
that doesn't mean that multiplication takes precedence over division.
Or addition takes precedence over subtraction.
See these last four we have to be careful, multiplication and division are actually on
the same level.
And so are addition and subtraction.
So if you see only these operations available, we'll simply do them from left to right.
Uh, and that's, that's the case with both of these guys.
And so this again helps us eliminate how many parenthesis we have.
So ah, for instance, if we had ten minus three, times two, there's no more ambiguity.
I don't have to clarify whether I mean ten minus three, times two, or if I mean ten minus
three, times two.
If there are no parenthesis listed, and there are no exponents, then multiplication takes
precedence over subtraction.
So I would combine the three and the two first to give a six.
And then we'll do ten minus six, which is four.
Okay, and this is consistent for whoever works the problem, if everybody follows the order
of operations, that will really cut down on how many parenthesis we have to write all
over the place.
Because everybody understands that we're going to do the operations in this order, unless
specified differently by parenthesis.
Okay, now how do you remember the order of operations?
Because it's, you know, you got exponents and subtraction, and division, and it's easy
to forget what order these are supposed to come in.
So most people when they teach order of operations use kind of a, a juvenile type of mnemonic
device.
It's a little silly, but it works really well, and a lot of people use it.
It's a sentence that says "Please Excuse My Dear Aunt Sally".
Where the leading letters tell you all you need to know.
Parenthesis, exponents, multiplication, division, addition, subtraction.
So just remember the sentence "Please Excuse My Dear Aunt Sally", and you'll be able to
remember the order of operations.
Alright let's close out with couple of quick examples.
If you had three times four, minus two, because we know "Please Excuse My Dear Aunt Sally",
then I know that multiplication, this operation, takes precedence over subtraction, which is
this operation.
And so the order, the kind of the default order, the understood order, that these operations
should be done, would be to do three times four first, which is twelve.
Then we'll subtract two from twelve, and get ten, so that would be our answer.
But these become especially helpful for really big operations like this.
Here we have seven plus five, divided by four, minus three, times nine, plus one, plus seven.
This could be interpreted, if we didn't have order of operations, a dozen different ways.
And so, ah, before order of operations somebody would have to literally group this, you know,
literally, you know, every which way, uhm, just so everything is clear, what order we're
supposed to be doing these in pair wise, or two, two numbers at a time.
And, so we don't want to have that many parenthesis.
So if we know, Please Excuse My Dear Aunt Sally, then I'm going to do all the parenthesis
first.
Which there's a parenthesis here, which is nine plus one.
And even though it's not listed, there's actually a parenthesis here as well.
Uhm, any numerator or denominator is its own set of parenthesis.
And the reason that is, is that when you see division, what would you divide four by, or
four into?
What would we divide by four?
Uhm, I don't know, it's seven plus five, whatever that is, and so obviously we would have to
figure out what seven plus five is, before we could divide it by four.
And so there's an understood grouping in any numerator or denominator.
So step one would be to say twelve divided by four, minus three, times nine, plus one,
makes ten.
And then minus seven.
Okay, so parenthesis are done.
There are no exponents, next we'll do multiplication, division and there on the same level, uhm,
so we'll do them all at the same time.
So twelve divided by four is three.
Three times, ten makes thirty, and we still have minus seven on the end.
But I'm not doing the subtraction yet.
Oh, I'm sorry, and this is addition, as a type of, my mistake there, that should be
a plus.
Three times ten, plus seven.
Just a typo.
Alright, and then, lastly we have addition and subtraction.
Now here's where you also have to be careful, somebody might misunderstand me and say, oh
well, addition comes before subtraction, so I better combine these first.
That's not the case.
If you look back at what I said earlier addition and subtraction are on the same level so these
operations are done from left to right if that's all you have.
Okay, so addition does not take precedence over subtraction.
So we'll do three minus thirty, that's negative twenty-seven.
And negative twenty-seven, plus seven, makes negative twenty.
So anyways, so you see how helpful, or how extremely helpful order of operations can
be, because they limit how ugly our expressions are looking, because of eliminating the need
for tons and tons of parenthesis all over the place.