### 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.