Hey, guys.
Welcome to Algebra 1.
Today we're going
to get started
solving problems
about a power of a product.
Use all you know about
exponents and patterns
because they're going to help
you get through this lesson.
Ready to get started?

(Describer) She uses a stylus on a screen.

Let's go.
All right,
to help you with this
I need to set up
a pattern for you.
Bear with me,
but it's going to help you
understand how you solve
these kind of problems.
4Y raised to the 3rd power,
that product.
What does that actually mean
if we expand it out?
I'm multiplying 4Y
by itself three times.
So 4Y times 4Y...
times 4Y.
Right?
The commutative property
of multiplication
tells me I can
multiply things
in any order
that I feel like, right?
I'm going to rearrange
this multiplication.
I'll bring all those
coefficients out front--
all those 4's--and I'll
group all those Y's together.
Okay? I'm going to represent
this as 4 times 4 times 4
and Y times Y times Y.
Right?
Okay, now I'm going
to simplify this.
I see this part, right here.
I've got those three 4's
multiplied together.
Then here I've got those
three Y's multiplied together.
I could represent
that first chunk,
that 4 times 4 times 4,
as 4 to the 3rd
and I could represent
that second chunk,
Y times Y times Y,
as Y to the 3rd.
That means that 4Y
raised to the 3rd power,
it means the same thing
as 4 to the 3rd
times Y to the 3rd, right?
Okay, keep that in mind
and keep going with me.
Do this a few more times.
So this one's
a little different,
but we'll apply
our same rule,
that same process
that we just did.
So 10B cubed
raised to the 2nd power.
That means I'm multiplying
10B cubed by itself
just one time.
I would have 10B cubed...
times 10B cubed.
Right?
If I expanded that out, okay?
I'm going to rearrange things
like we did before.
I'll bring my 10s out front
and put my B cubes
to the back.
I'm going to rearrange this
as 10 times 10
times B to the 3rd
times B to the 3rd, right?
Just rearranged the order
of the multiplication.
Didn't change the meaning
of the problem, okay?
Now I'm going
to look at these in groups.
I'll throw those parentheses
back in there
to simplify this.
If I'm multiplying
10 by itself one time,
that's the same thing
as 10 squared
because you just raise
10 to the 2nd power.
Multiply it by itself once.
Then over here
I have B to the 3rd
times B to the 3rd, right?
Isn't that the same thing
as squaring B to the 3rd,
just multiplying it
by itself one time?
I could represent it
like this:
I've got 10 times 10,
which I'm representing
as 10 squared,
and I've got B cubed
times B cubed,
which I'm representing
as B cubed squared.
That means
that 10B to the 3rd
raised to the 2nd power,
that problem
we started with,
could be written
just as this.
10 squared times B cubed
to the 3rd, okay?

(Describer) B-cubed squared.

Stay with me,
keep going.
3X squared,
that whole quantity,
raised to the 4th power.
All right, so let's think about
what this means.
You're multiplying 3X squared
by itself 4 times.
So you would have
3X squared...
times 3X squared...
times 3X squared...
times 3X squared.
Okay?
Now I'll do
that same rearranging.
I'll bring
all those 3's out front
and put all those variables
at the back.
All right, so I've got four 3's
I'm multiplying together,
so one, two, three, four.
Now, I pulled
each of those 3's out front.
And then I have X squared
that I'm multiplying
by itself four times.
So I'll bring all those
together at the back.
Okay, now I'm going
to wrap parentheses
around those different groups
just to help me
zone in on them.
All right, so I have
3 times 3 times 3 times 3.
I could represent that
as 3 to the 4th, right?
Because you're multiplying 3
by itself 4 times.
Now, here you have X squared,
times X squared,
times X squared,
times X squared.
You're multiplying X squared
by itself four 4 times.
So you could represent
that part as X squared,
that quantity,
raised to the 4th power.
That means that 3X squared,
that whole bit,
raised to the 4th power
is the same
as 3 to the 4th
times X squared to the 4th.
Are you noticing
anything yet?
Let's keep going.
All right, let's look at
those three answers together,
and see if you notice
the pattern.
See if you notice
what's going on.
Okay, when we did 4Y,
that quantity cubed,
we ended up simplifying it
as 4 to the 3rd
times Y to the 3rd.
Right?
And when we had 10B cubed
raised to the 2nd power,
we represented that
as 10 squared times...
Let me write it
just like we wrote it.
I want you to really look
at them all in the same way.
Okay. I'm going to write
10 squared times--
I've got to do B cubed squared.
I was trying it with the
parentheses, but I can't.
And then we had 3X squared
raised to the 4th power,
right?
We had 3 to the 4th
and X squared to the 4th.
Now look at those
all together,
even pause me for a bit.
See if you notice
the pattern.
There's something going on
with the exponents
that we start with
in our original problem
and the exponents that
we end up with in the solution.
Pause me for a bit.
See if you can figure out
that pattern.
All right,
did you notice?
Let's look at it together.
In that first one,
we had 4Y
raised to the 3rd power,
and then for our answer
we had 4 to the 3rd
times Y to the 3rd.
We distributed that exponent
throughout that
product inside.
We broke it up,
raised 4 to the 3rd power
and then raised that Y
to the 3rd power.
If you look
at that second example,
that 10B to the 3rd squared,
it's like we took that 10
and raised it to the 2nd power
and then we took that B cubed
and raised it
to the 2nd power.
It's like we took
both parts and raised them
to the power of our exponent
on the outside.
Look at that last one.
3X squared
raised to the 4th power.
We took that 3 and raised it
to the 4th power
for this part,
and we took that X squared,
and raised it
to the 4th power
for this part.
It's like you're distributing
your exponent
throughout the terms
inside your parentheses.
That rule is known as...
the power of a product.
If you have a product that you
have to raise to some power,
just split the factors
of your product up
and raise them
to that power initially.
That's a shortcut
to solving these problems
that involve
a power of a product.
Let's apply this rule
and solve a few more.

(Describer) Titles: Power of a Product.
Open parenthesis a-m close parenthesis, to the r power, equals a to the r power times m to the r power.

(female narrator)
Power of a Product:

(AM) to the R power
equals A to the R power
times M to the R power.
All right, let's get
that pen back.
Okay.
I have 5B cubed C,
that quantity,
and I'm raising it
to the 2nd power.
Instead of expanding it out
and finding the pattern,
I'll use the rule
we just figured out.
I'll take each of these terms,
each of those bits,
separately,
and I'll raise them
to the 2nd power.
All right?
Watch me work.
I'm going to take that 5
and raise it to the 2nd power,
so 5 squared, then I'm going
to take this B cubed
and raise it
to the 2nd power.
So B cubed, squared.
Then I'm going
to take that C
and raise it
to the 2nd power, okay?
All right,
so now let's simplify this.
Now that you know the rule,
we're going to go ahead
and simplify 5 squared, okay?
We know that 5 times 5
is what this represents, right?
So 5 squared,
that's 5 times 5.
So that's 25, right?
I know that that first part
just simplifies to 25.
Now for this one, that B cubed
raised to the 2nd power,
let's come off to the side.
B cubed raised
to the 2nd power.
We'll use our rule about
raising a power to a power.
Remember when you raise
a power to a power,
you just multiply
your exponents together.
I know 3 times 2, that's 6.
That means that that simplifies
as B to the 6th, right?
I know the second part,
right here,
I can rewrite
as B to the 6th.
You see I'm dropping my dots?
I'm getting ready
to write the answer.
You could keep writing those
multiplication dots, though.
Then I see C
raised to the 2nd power.
I'll come off to the side.
So C raised
to the 2nd power.
Did not write that
like it was.
Let's write it
just like we started.
Okay, so C
raised to the 2nd power.
Okay, whenever you don't see
an exponent,
remember there's a 1 there,
it's an invisible 1.
Same thing,
power to a power,
it's like you're multiplying
your exponents together,
that is what you're doing.
So 1 times 2 is 2.
So C raised to the 2nd power
is C squared.
You probably already
knew that anyway, okay?
Then you're all done.
That would break
right on down
to 25B to the 6th,
C to the 2nd power, okay?
Let's try another one
just to make sure
you got the hang of it.
So for this one, we have
4X to the 6th, Y squared,
raised to the 3rd power,
right?
Just use our rule.
We're going to raise
each of these parts
to the 3rd power,
like we're distributing
the exponent, almost.
All right, so I'd have 4
to the 3rd power.
I'm going to raise everything
to that exponent
on the outside.
That's times X to the 6th,
raised to the 3rd power,
right?
Then I'd have Y squared
raised to the 3rd power.
Now I'm just
going to come off to the side
and do a little work
to simplify all this.
All right.
So 4 to the 3rd power.
I know that that's 4
times 4, times 4.
So 4 times 4,
that's 16.
And 16 times 4 is 64.
If you don't want
to do mental math
like I'm doing right now,
you could pull out a calculator
and do 4 to the 3rd
and get 64.
We've got that first part.
Now we have X to the 6th
raised to the 3rd power.
We've got to handle this.
Remember, we use our rule
about applying
a power to a power.
Just multiply
your exponents together.
So 6 times 3, that's 18.
That would be X to the 18th.
And then I have--
I'll write over here,
I'm running out of room--
Y squared
raised to the 3rd power.
So our rule about
raising a power to a power,
just multiply
your exponents together.
So 2 times 3, that's 6.
This would be Y to the 6th,
for that last bit.
And you're all done.

(Describer) 64x to the 18th, y to the 6th.

(female narrator)
64X to the 18th, Y to the 6th.
Just distribute
that exponent throughout
and you're all done.
All right.
Okay, one more together.
So I have 2MN to the 5th
raised to the 4th power.
So I'm going to just
distribute that exponent--
I'm thinking about it
like that--throughout.
So I have 2
raised to the 4th power.
That's 2 to the 4th.
Then I have M
raised to the 4th power.
So times M to the 4th.
And I have N
raised to the 5th power--
I'm sorry, N to the 5th
raised to the 4th power.
N to the 5th,
raised to the 4th.
All right,
now let's simplify this.
So I'm going
to come off to the side.
2 to the 4th,
that's 2 times 2,
times 2, times 2,
right?
So 2 times 2 is 4.
4 times 2 is 8.
8 times 2 is 16.
Okay?
So I've simplified
that part to 16.
M to the 4th.
I did a shortcut there.
I did it differently
than that one
that had a 1 inside
for the exponent.
It'll save me work.
It's just M to the 4th.
There's a 1 there
I don't see,
so you're not
changing anything
when you raise that
to the 4th power.
Let me just show you,
just in case you need it.
There's a 1 there
that I don't see.
So when I multiply
1 times 4, it's just 4.
You're raising it to that
exponent on the outside,
not multiplying it
by a number
that will change
that exponent's value.
Give yourself a shortcut
when you run into that stuff.
For this one,
I have N to the 5th
raised to the 4th power.
Come off to the side.
So N to the 5th
to the 4th, right?
We're raising
a power to a power.
Just multiply
your exponents together.
So 5 times 4, that's 20.
So N to the 20th,
for the last bit.
And you're all done.
You handled
all three pieces of that.
Okay?
All right, let's keep going.
It is now your turn, so pause,
take a few minutes,
and work your way
through these problems.
To compare answers with me,
press play.

(Describer) Titles: Simplify the following expressions.
Number One: 6g to the fourth, h to the third, all squared.
Number Two: 3d to the 11th, p to the 6th, all to the third.

(female narrator)
Simplify the following
expressions.
Number 1: 6G to the 4th,
H to the 3rd, all squared.
Number 2: 3D to the 11th,
P to the 6th,
all to the 3rd.

(instructor)
All right,
let's see how you did.
6G to the 4th, H to the 3rd,
that quantity,
raised to the 2nd power.
You should have got 36G
to the 8th, H to the 6th.
3D to the 11th, P to the 6th,
that quantity cubed.
You should have got 27D
to the 33rd,
P to the 18th, okay?
If you need to see how I did
any of those,
let me show you.
All right,
so that first one.
6G to the 4th H cubed,
that quantity
raised to the 2nd power.
Make sure I got my pen.
I'm going to distribute
that exponent
throughout that product
in my parentheses.
I'm going to raise 6
to the 2nd power,
I'm going to raise G
to the 4th,
to the 2nd power.
Let's get that
multiplication in there
to break things up.
And I'm going to raise H cubed
to the 2nd power.
Right? So then I'm going
to come off to the side.
I need to get 6 squared.
Okay, 6 squared,
that's 6 times 6.
So that's 36.
Okay? I'm going to replace
6 squared with 36.
Then I have this piece,
that G to the 4th,
raised to the 2nd power.
So G to the 4th,
raised to the 2nd power.
You're going to multiply
your exponents together
because you're
applying your rule
about raising
a power to a power.
Okay, so 4 times 2,
that's 8.
So you have G to the 8th,
and then our last bit,
that H cubed
raised to the 2nd power.
Same thing;
apply your rule
about raising
a power to a power.
Multiply your
exponents together.
3 times 2, that's 6.
So H to the 6th
for the last bit, okay?
That is how I got
that first one.
All right, if you need
to see the second one,
stay with me.
Here we go.
I'm going to distribute
that exponent throughout
what's inside my parentheses,
basically what I'm doing.
All right, so I have 3
raised to the 3rd power
times D to the 11th,
raised to the 3rd power
times P to the 6th,
raised to the 3rd power.
Okay?
Then just simplify this.
Just keep breaking this down.
So 3 to the 3rd--
come off to the side.
All right,
so that's 3 times 3 times 3.
So 3 times 3, that's 9.
9 times 3 is 27.
So I know 3 cubed
would simplify to 27.
Then I have D to the 11th,
raised to the 3rd power.
Come off to the side.
D to the 11th,
raised to the 3rd power.
Follow your rule about
raising a power to a power.
Multiply your
exponents together.
So 11 times 3, that's 33.
So that would be D
to the 33rd power.
Then we have our last piece,
that P to the 6th
raised to the 3rd power.
Multiply your
exponents together.
So 6 times 3, that's 18.
So This would be P
to the 18th.
And that's the last piece.
All right,
that is how I got that answer.
All right.
I hope you got a handle
on the problems
we did today,
and that you saw
how using your knowledge
of exponents and patterns
can help with the laws
of exponents.
All right,
see you next time.

(Describer) Accessibility provided by the US Department of Education.