>> Math is cool and you can do it.
We're going to go onto part two
of evaluating algebraic expressions.
The difference is these are a little bit more involved
and will be having some negative numbers.
So we're going to evaluate several different algebraic
expressions using variables A, B,
and C and where is negative 2,
B is negative 3, and C is negative 1.
So here's the first one.
A squared minus, and in parentheses, B plus C. So,
we need to replace the variables with the given numbers
and then use the order of operations.
Now, any time you have a variable raised to an exponent,
what you could do is write that as a multiplication,
or you could leave it in parentheses.
So there is actually a couple
of ways you might start this problem.
You could instead write it
as A times A minus B plus C before replacing the variables,
for instance, all right.
But let's go ahead and just replace the variables over here.
So if I put A squared, I must put negative 2 in parentheses,
because that's what A really is, otherwise,
I would only be squaring the 2.
And we want to make sure that it says A is being squared.
So that any time you replace the variable with a negative number
and that number is being raised to some power, in other words,
the variable goes to some power,
you absolutely must use parentheses.
Okay. And we're going to just copy the rest of the products,
the minus sign, the parentheses, plugging in negative 3 for B
and negative 1 for C. Okay.
And now let's go ahead and do the order of operations.
We need to do what's inside the parentheses,
but we can also square this number at the same time
because the last thing we'll do is the subtraction
in the middle.
So, negative 2 to the second power, remember what that means?
It's negative 2 times negative 2, so that's 4 minus,
and then in parentheses here,
negative 3 plus negative 1 is negative 4.
And, of course, now, subtracting negative 4 is the same
as adding a positive 4.
So our answer to this problem is 8.
You might have taken one more step by writing
out this negative 2 squared as negative 2 times negative 2.
I'm actually going to show you how --
that's how it will over here if I did it --
if before replacing the variables I wrote it
as A times A, then I can plug in the negative 2.
And you can put it in parentheses
to show multiplication, or you could just put a dot
between the two negative 2s.
And we're going to plug in for C negative 3 plus negative 1,
B plus C, and then proceed just like we did on the left.
We're still going to get 4 minus a negative 4,
which is 4 plus 4, or 8.
Again. So there's a couple of ways you might want to write it.
I suggest doing it the second way
where you write A times A. Some people write this wrong.
They put a negative 2 squared and they don't remember
to put the negative 2 in parentheses,
so the first step would be wrong.
Here's another one.
If you want, put it on pause and try it on your own first.
Okay. So what are we going to do with this problem?
We're going to replace B with negative 2, and notice,
there's only one variable: a B. It's in two different places,
but there's only one variable.
So we won't worry with a negative 2 or the negative 1.
So again, you can write that as B times B,
and then you could write this as 5 times B, or,
you could have just replaced the B in right away with negative 3,
but you'd have to put it in parentheses again.
And then negative 5 times negative 3 plus 11.
So you could write it the way I id over here on the right,
or we could do the replacement now.
This is negative 3 times negative 3 minus 4 times
negative 3 plus 11.
So what we're doing is replacing B with negative 3.
All right, now we're going to do order of operations,
so I need to do multiplication next.
And there's two places there's multiplication.
I'm using the order of operations.
So negative 3 times negative 3 is 9.
This will be minus, right, there's the minus sign,
and then 5 times negative 3 is negative 15.
You could put that in parentheses
or just write negative 15 plus 11.
And remember, minus a negative 15 is the same
as plus positive 15.
And then we simply add those three numbers.
There are three positive numbers.
I'm going to add the 9 and 11 first to get 20 plus 15,
and that will give me 35.
Over on the right, if you would have just simply replaced the B
with a negative 3 to begin with, everything,
the next step will look exactly like that.
9 minus negative 15 plus 11.
Okay, let's going on to another problem.
Okay, this time we're going to have to replace something
in for all three different variables, so let's go ahead.
So we're going to copy the problem but replace the B
with negative 3, the C with negative 1,
and the A with negative 2,
so I've done the first fraction here.
And then we're going to have to write the parentheses.
And negative 3 for B plus 1 squared.
Now remember, in order of operations you have
to simplify inside the parentheses first.
You do not say that this is 9 plus 1.
You don't just square each number and add it;
that's not going to work.
You have to actually add inside the parentheses,
and then square the number.
Also in order of operations when you have a fraction,
remember you have to think it's as if the numerator
and denominator were in parentheses,
they have to be simplified first before reducing the fraction
or just dividing.
So in the numerator, this fraction I have
to add the negative 3 plus negative 1 which is negative 4.
And the second part over here I need
to add inside the parentheses which is a negative 2.
Now you must keep that negative 2 in parentheses
because you're going to square it, so it's absolutely necessary
to still write the parentheses.
Okay. The last thing we're going to do is add, right.
So I could independently do this division
and square this number at the same time.
So negative 4 divided by negative 2 --
two negatives when you divide it will be a positive.
4 divided by 2 plus -- and remember what this means.
Negative 2 times negative 2 is 4.
So, I have 2 plus 4, and the answer is 6.
All right.
Next one. Okay.
Make sure I'm going to right it over to the right.
3C cubed. What does that mean?
I'm going to write out what it means just to remind you.
3 is not being cubed, only the C is being cubed,
so it really means 3 times C times C times C. Just
as a little reminder.
So now, I'm going to plug them in.
So I have 3 times negative 1 times negative 1 times
negative 1.
That's the value for C, and B is negative 3.
So let's do the numerator and the denominator.
In the numerator I'm just multiplying the 3 times 1 times
1 which is 3, and then I have
to decide whether the answer is positive or negative.
Well, three negatives means there's going
to be one negative.
Now what happened here in the denominator?
Negative 3 plus 3 -- that's zero.
Well, that's not simplified.
You never have an answer simplified if you have a zero
in the numerator or the denominator.
What happens?
What is negative 3 divided by zero?
That's undefined.
You can't leave your answer with a zero in the denominator;
you have to say that it's undefined.
Again, if you want, you can write out what this means.
2A cubed is 2 times A times A times A. 5B squared is 5 times B
times B. And 6C to the eighth is six times --
then you'd have to write eight Cs multiplied together.
I don't think there's room on this page.
I'm going to do a little bit modification here.
So I'm going to have 2 times A to the third power.
I am going to write the A down three times, but I'm going
to do replacing it with a negative 2.
So I'm going to have negative 2 times negative 2 times
negative 2.
So there's your 2 times your A cubed minus 5 times your
B squared.
Okay, well that's a negative 3 times a negative 3.
And now, I have 6 times C to the 8.
I don't have room to write eight of those,
so for the moment I'm just going to write down what that means.
It's negative 1 to the eighth power, and we're going
to figure that out in a minute.
So what do we have here?
2 times negative 2 times negative 2 times negative 2.
So basically I have 4 times 4, which is 16.
Three negative signs multiplied together is a negative,
minus -- now be careful here.
You have this minus sign,
and then I'm multiplying 5 times negative 3 times negative 3.
So, the negative 3 times negative 3 is 9,
so I have 9 times 5, so that's 45.
Okay. Plus 6 times - remember, I have to do exponents first?
So let's look at this.
What's going on.
I'm doing a negative 1 to the eighth power.
So, we know we're going to have eight negative ones in a row,
and all those ones times each other are going to be one,
and if I have eight negative signs,
will the answer be positive or negative?
It will be positive 1.
So what do I have here?
Negative 16 minus 45 plus 6.
I'm going to add the negative 16 and 6 together first,
because that's easy, negative 10 minus 45, and last,
negative 10 minus 45 is negative 55, and you are done.
I hope you learned something.
Remember, math is cool.