Order of Operations

Let's do some order of
operations problems, and for

the sake of time I'll do
every other problem.

So let's start with 1b.

1b right there.

They have 2 plus 7 times 11
minus 12 divided by 3.

So just remember, the top
priority is always going to be

your parentheses.

So you have your parentheses--
Let me write it this way.

Your top priority's going to
be your parentheses, after

that you're going to have your
exponents, after that you have

multiplying and dividing, and
after that you have addition

and subtraction.

So let's remember that and
tackle these order of

operations problems.

So priority, there's no
parentheses here, there's no

exponents, so the priority's
going to go to multiplication

and division.

So you could view this as being
equivalent to-- So we're

going to do our multiplication
before we do any addition or

subtraction, and we're going
to do our division before

doing any addition
or subtraction.

Problem 1b is exactly equivalent
to this, the

parentheses are just-- I'm
reinforcing the notion that

I'm going to do my
multiplication and division

before I do the addition
and the subtraction.

So 7 times 11 is 77, and then
12 divided by 3 is 4.

And the rest of the problem was
2 plus this thing, which

is 77, minus this thing.

And here, since everything is
in addition or subtraction,

let's just go left to right.

2 plus 77 is 79 minus 4,
which is equal to 75.

So 1b is equal to 75.

Let's do 1d.

This is a nice hairy problem
right there.

So 1d.

It says 2 times 3
plus 2 minus 1.

Closing two parentheses, all of
that over 4 minus 6 plus 2

minus 3 minus 5.

Let's see if we can simplify
this a little bit.

As we said, parentheses
take our priority.

So let's do the parentheses
first. 2 minus 1.

2 minus 1 is just 1.

3 minus 5.

That is minus 2, or negative
2, I should say.

6 plus 2 is 8.

Now let's keep looking at the
parentheses to see where we

can simplify things.

We have this parentheses
right here.

So 3 plus this 1 is now going
to be equal to 4.

Actually, let me rewrite it.

So we're going to have 2 times
this whole expression, 3 plus

1, so it's 2 times 4.

That right there is 4.

All of that over 4 minus
8, that's negative 4.

This right here is negative 4.

And then minus this
negative 2.

So minus negative 2.

2 times 4 is 8, so this whole
thing simplifies to-- A minus

of a negative, that's just
the plus of the plus, the

negatives cancel out.

So this whole thing simplifies
to 8 divided by negative 4 is

negative 2 plus 2.

So it equals 0.

So this big, hairy thing
simplified to 0.

Now let's do 2b.

Let me clear some space here.

I'll leave the order of
operations stuff there.

Let me clear that and
let me clear this.

All right, 2b.

Evaluate the following

expressions involving variables.

Fair enough.

So they wrote 2y squared, and
they're saying that x is equal

to 1, which is irrelevant
because there is no x here,

and y is equal to 5.

So if y is equal to 5, this
thing becomes the same thing

as 2 times 5 squared.

And notice, I put parentheses
there.

I could have written this as,
this is the same thing as 2

times 5 squared.

And if you look at the order of
operations, exponents take

priority over multiplication.

That's why in my head I just
automatically put those

parentheses.

We're going to do the
exponent first.

So this is 25, so you get 2
times 25 is equal to 50.

So that is 2b, this is equal
to-- use a darker color --that

is equal to 50.

Let's do 2d.

They're giving us y squared
minus x, whole thing squared.

x is equal to 2 and
y is equal to 1.

Well, we just substitute.

Where we see a y we put a 1.

So this is going to be 1 squared
minus x squared--

Sorry, minus x, not x squared.

So we just put a regular
x there.

That's where we put a 2.

And then all of that squared.

Well 1 squared is just
1, so that is just 1.

1 minus 2 is negative 1.

And then we're going to want to
square our negative 1, so

that will be equal
to positive 1.

So that is equal to 1.

Negative times a negative
is a positive.

All right, let's do 3b.

Doing every other problem.

I'll do it in yellow.

Evaluate the following

expressions involving variables.

All right.

Same idea.

So they're giving us
4x over 9x squared.

Oh, actually I said I'd
do 3b, I was doing 3a.

So here we go.

We have z squared over x
plus y plus x squared

over x minus y.

And they're telling us that x
is equal to 1, y is equal to

negative 2, and z
is equal to 4.

So let's just do our
substitutions first.

So z squared, that's the same
thing as-- I'll do it in a

different color --4 squared over
x, 1, plus y, negative 2,

plus x squared, that's
1 squared, over x,

which is 1, minus y.

y is negative 2.

So this is going to be equal to
4 squared is 16 over 1 plus

negative 2, that's 1 minus 2--
it's just a negative 1 --plus

1 squared, which is 1, over
1 minus negative 2.

That's the same thing
as 1 plus 2.

So it's 1/3.

And so this will be 16 divided
by negative 1.

We could write that as that's
equal to negative 16 plus 1/3.

Now if we want to actually add
these as fractions we could

have a common denominator.

Negative 16 is the same thing
as minus 48 over 3, or

negative 48 over 3.

If you take 48 divided by 3
you'll get 16, and I'm just

keeping the negative sign.

And then you add
that plus 1/3.

We have a common denominator
now, 3.

Negative 48 plus 1
is negative 47.

So our answer is negative
47 over 3.

Problem 3d.

Same type of situation.

x squared minus z squared over
xz minus 2x times z minus x.

x is equal to negative
1, z is equal to 3.

Let's do our substitutions.

So this is x squared.

That's minus 1 squared.

Minus z squared, so
minus 3 squared.

All of that over x times z.

x times z is minus 1 times 3,
minus 2 times x, x is negative

1, times z minus x,
times 3 minus x.

x is negative 1 minus x.

Wherever we saw an x
we put a minus 1.

So this is going to be equal
to-- Remember, you do your

exponents first. Well,

parentheses first, then exponents.

So we have negative 1 squared,
that's just a positive 1.

3 squared, that's just
a positive 9.

So our numerator becomes 1 minus
9, that's minus 8 or

negative 8.

And then our denominator.

Negative 1 times 3
is negative 3.

And then let's go to our
parentheses here.

We have 3 minus negative 1,
that's the same thing as 3

plus plus 1.

So that right there becomes 4.

So our denominator becomes
negative 3 minus 2 times

negative 1 times 4, so
that's negative 8.

Minus negative 8.

Minus of a negative is the
same thing as a plus.

So this whole thing becomes
negative 8 over negative 3

plus 8 is 5.

So it's negative 8/5,
minus 8 over 5.

All right, let me clear up
some space just so we can

reference this problem
properly.

Let me clear all of this
out of the way.

These are interesting now.

Problem 4: insert parentheses in
each expression to make it

a true equation.

Fascinating.

So 4b.

You have 12 divided by 4 plus
10 minus 3 times 3 plus 7 is

equal to 11.

So let's see what happens if we
just do traditional order

of operations, and I'll do a
little bit in my head because

this is going to take some
experimentation.

Oh yeah, this is 4b,
12 divided by 4--

Yep, that's the problem.

So if did 12 divided by 4 first,
and we would get 3.

So let me just do
this in yellow.

So if we did regular order of
operations this would be a 3.

This right here would
it be a 9.

So you would have 3 plus 10,
which is 13, minus 9, 13 minus

9 is 4 plus 7.

Actually, that seems right.

Let me make sure I
did that right.

3 plus 10-- Right,
that looks right.

So we really just have to do
regular order of operations.

So it already looks like
a true equation.

So if you do 12 divided by 4
plus 10 minus 3 times 3 plus

7, I think it turns out right.

Let me confirm.

Make sure I'm not making
a mistake.

12 divided by 4 is 3 plus 10
minus 3 times 3 is 9 plus 7.

This is equal to 13 minus 9,
which is equal to-- So all of

this is equal to 13 minus 9 is
equal to 4 plus 7 is, indeed,

equal to 11.

So that one wasn't too bad.

You actually wouldn't have to
put any parentheses to make

this a true expression.

You would just have to follow
the order of operations.

But throwing those parentheses
there makes it a little bit

easier to read.

Let's try 4d.

12 minus 8 minus 4 times
5 is equal to minus 8.

So first let's just see what
happens if we did traditional

order of operations.

If we did traditional order of
operations we would do this 4

times 5 first, which would
give us 20 over there.

And then we would have
12 minus 8 is 4.

And then we would do 4 minus
20-- No, that's not right.

That would give us
negative 16.

So that's not going
to be right.

So we can't just do traditional
order of

operations.

Sorry, this is a minus
8 right there.

So let's see how we can
experiment with this.

Let's try out a couple
of situations.

What if we did 12 minus 8 minus
4 and then multiplied

that times 5.

Let's see what this give us.

I'm just experimenting
with parentheses.

So if you do 8 minus 4, that
right there would be

8 minus 4 is 4.

And then 4 times 5 would be
20, and then 12 minus 20--

yeah, that works.

So let me confirm that.

So I'm saying I'm going to put
parentheses right there and

right there and let's
work it out.

You would get 8 minus 4 is 4.

So this whole thing was
simplified to 12

minus 4 times 5.

And you just do order of
operations, you do

multiplication first.
So that is just 20.

And if I wanted to make it very
clear I could actually

write it like this.

I could actually put
another round of

parentheses right like that.

But order of operations would
tell us to do it anyway.

So it becomes 12 minus 20, which
is, indeed, minus 8 or

negative 8.