Hey guys,
welcome to Algebra 1.
Today we solve problems
involving the power
of a quotient.
Your knowledge of exponents
and products and quotients
will take you far
in this lesson.
Ready to get going?
Let's start.
All right,
let's get the pen here.
Okay, knowing what you do about
exponents, let's expand this:
7 halves to the 3rd power.
I'm raising 7 halves, that
quotient, to the 3rd power.
That means I'm multiplying
it by itself 3 times.
I could expand this
to 7 halves times 7 halves
times 7 halves, right?
Just expand it out--
what exactly that means.
Now to make it simpler,
I'm going to just put
all of those 7s together
for my numerator--
7 times 7 times 7.
And I'm going to do the same
thing with my denominator:
2 times 2 times 2, right?
Didn't change
the meaning at all.
Just representing it
differently.
Now if I wanted to simplify this
and write it and represent it
a different way,
I know that 7 times 7 times 7
is the same
as 7 to the 3rd,
and that 2 times 2 times 2
is the same as 2 to the 3rd.
That means that 7 halves
to the 3rd power
is the same as 7 to the 3rd
divided by 2 to the 3rd.
Okay, keep going.
We establish a pattern
that's going to help us
understand this rule.
So 4/9 to the 5th--
I'm going to use that same
process to understand this one.
So 4/9 to the 5th power,
that's the same as
4/9 times 4/9
times 4/9 times 4/9 times 4/9.
Right?
the product of 5 "4/9ths."
Now I'm going to group
those numerators together.
Let me get
one big fraction bar.
I write that as 4 times 4
times 4 times 4 times 4, right?
I got the product
of those 5 4s
together with my numerator,
and same
for the denominator:
9 times 9 times 9
times 9 times 9, right?
Now if I wanted to simplify that
and bring my exponents back,
I have the product
of 5 4s in my numerator.
That's 4 to the 5th up top.
And I have the product
of 5 9s
in the denominator.
That's 9 to the 5th
on the bottom.
So 4/9 to the 5th, the problem
we originally started with,
could be written as
4 to the 5th over 9 to the 5th.
You starting to notice
the pattern yet?
Let's try another one.
All right,
c over d to the 4th power.
Applying that same process,
I know I could expand this
and write it as:
c over d times c over d
times c over d times c over d.
Right, okay.
Then I'm going to
group all those cs together.
So I'd have c times c
times c times c
over d times d
times d times d, right?
Then I'm going
to represent that
using exponents
to write it simpler.
The product of 4 cs
in my numerator,
that's c to the 4th.
And the product of those 4 ds
in the denominator,
that's d
to the 4th, right?
That c over d to the 4th power
could be written
as c to the 4th
over d to the 4th.
Do you see
what's going on?
It's really like you're
distributing your exponent
throughout the terms of
what's inside your parentheses.
In this problem, c over d raised
to the 4th power means the same
as c to the 4th power
over d to the 4th power.
That's the rule.
Let's look at it
written algebraically.
Get that
pointer tool back, okay.
So for the power
of a quotient,
anytime we're
dividing a quotient
and we're raising it
to the power,
you can split it.
Raise your numerator
to whatever your power is
and raise your denominator
to whatever your power is.
It's like
you're distributing
the exponent throughout
your quotient.

(female describer)
Title: Power of a quotient--
a over m to the r power
equals a to the r power
over m to the r power.
Let's apply this;
let's try a couple of problems.
Our first example, we have x
over 8 raised to the 2nd power.
I'm going to apply that rule--
get the pen back--
for the power of a quotient.
I raise my numerator
to the 2nd power.
That would become
just x to the 2nd power.
I'm going to raise my
denominator to the 2nd power--
just 8 to the 2nd power.
Because I don't know what x is,
I can't make that value
any more simple,
but I can represent
8 squared as 64, okay?
That means that your final
answer for this problem
would be
x squared over 64.
We need to raise
our numerator
and then raise
our denominator
to whatever that power is,
and you've applied the rule.
Okay, let's try another one.
Let's click off of that.
It's going to start
moving it around if I don't.
So p over 4 to the 3rd power--
so apply that rule.
Raise your numerator
to the 3rd power.
So p to the 3rd power.
And I'm going to raise
my denominator to the 3rd power,
so 4 to the 3rd power, okay?
I don't know what p is.
I can't represent that
another way.
It's just p to the 3rd.
I can't represent it simpler.
Now 4 to the 3rd--
Let's come off to the side.
I'm doing mental math,
but you can use a calculator.
You can get it that way.
4 to the 3rd;
that's 4 times 4 times 4.
So 4 times 4, that's 16.
And then 16
times 4, that's 64.
I'm going to put 64
in my denominator,
and I'm all done.
The answer to that problem
is p to the 3rd over 64.
All right,
now it's your turn.
Click off for a second;
there we go.
Take a look at these.
Pause the tape and try them.
See how you do.
When you're ready to compare
answers, press play.

(female describer)
Title: Simplify.
1. y over 9 to the 2nd power.
2. b over 2 to the 4th power.

(teacher)
All right, ready to check?
Here we go.
For the first one,
y over 9 to the 2nd power.
You should have
y squared over 81.
And b over 2
to the 4th power,
that simplifies to:
b to the 4th over 16.
If you want to see
how I did those, stay with me.
Let me get
my pen back, okay.
When I did the first one,
I just applied that rule.
I raised y to the 2nd power
and I raised
9 to the 2nd power, all right?
So, y squared--
couldn't do any more with that.
And then 9 squared--
I know that's 9 times 9,
so that's 81.
That's how I got
that first one, okay?
Now let's check
that second one.
Here I had
b over 2 to the 4th power.
I raised b to the 4th
and I raised 2 to the 4th.
I couldn't do anything
with that b to the 4th.
That stayed as it was.
Now 2 to the 4th--
and come off to the side.
That's the same
as 2 times 2 times 2 times 2.
I know 2 times 2 is 4,
4 times 2 is 8,
and 8 times 2 is 16.
That's how I got
that denominator,
and then I was all done.
All righty.
Now those were
some basic ones.
We're going
to step it up,
but we're going to apply
the same rule, okay?
Take a look at this.
It looks overwhelming,
but we'll break it up.
We apply the same rule.

(female describer)
3x to the 7th power, y squared
over 5z
to the 4th power, all cubed.

(teacher)
I see I have my numerator,
that 3x to the 7th,
y squared over my denominator,
that 5z to the 4th.
I'm raising that quotient
to the 3rd power.
I'm going to take extra steps
to break it down further
to make sure you're completely
clear with everything.
Okay?
Let's break this up.
I'm going to raise my numerator
to the 3rd power, right?
We know from what
we were just doing
when we apply that power
of a quotient rule, that's--
I could
expand it this way.
I raise that to the 3rd.
And the same
to my denominator.
I'm going to raise my
denominator to that same power.
With me so far?
Broke it up.
We raise our numerator
to the 3rd power,
and we need to raise
our denominator
to the 3rd power.
Now--
Now you apply a different rule
from your rules of exponents.
Now I have a product
that I need to raise to a power.
Remember, when you do that,
you raise each individual piece
to that 3rd power.
It's all we need to do.
I'm going to raise
this 3 to the 3rd power.
I'm going to raise
that x to the 7th
to the 3rd power...
and I'm going to raise
that y squared
to the 3rd power.
Okay?
All I did there was take each
individual piece in my numerator
and raise it
to the 3rd power.
I do the same thing
with the denominator.
I need to raise 5 to
the 3rd power,
and then I need to raise z
to the 4th to the 3rd power.
Okay?
Now that I'm looking at it
like this,
I need to keep simplifying
and see what this actually
ends up coming out to.
So, 3 to the 3rd--
I know that's 3
times 3 times 3.
3 times 3, that's 9.
And 9 times 3 is 27.
I know that
3 to the 3rd is 27.
Look at this x to the 7th.
Remember your rule about
raising a power to a power?
All you do is multiply
those exponents together.
That's the shortcut.
If I raise x to the 7th
to the 3rd power,
I just need to multiply
those exponents together.
7 times 3, that's 21.
That will become
x to the 21st.
The same thing here; I'm raising
a power to another power.
I need to multiply
these exponents together--
2 times 3...
That's 6.
So y to the 6th, okay?
Follow that same process
with your denominator.
You have 5 to the 3rd, right?
So 5 times 5, that's 25.
25 times 5, that's 125.
I have 125.
I have z to the 4th.
I'm raising that
to the 3rd power,
so my rule, raising
a power to a power--
I multiply exponents together.
4 times 3, that's 12.
So, z to the 12th power.
You can take it no further.
You are done, okay?
That's how you do that.
It's more complicated,
but you'll get it.
We'll do more.
Just break it up.
And you may not want
that middle step
because you'll get comfortable,
but I want to do it
to make sure
you get comfortable.
Break up your quotient,
raise it to whatever power
you're being asked to,
and apply the rule of exponent
you need to simplify it.
Let's try another one.
Let's keep going.
Let's make sure
I have enough room here.
Let's scoot that
over a bit, okay?
We have 9s squared,
t to the 5th over 4p cubed
q to the 8th, all raised
to the 2nd power.
All right.
I break it up first.
I split up my numerator
and my denominator.
I know that I need
to raise 9s squared,
t to the 5th to the 2nd power,
and I need raise 4p to the 3rd,
q to the 8th to the 2nd power.
Right? Okay.
I have to apply my rule about
raising a product to a power.
I raise each individual piece
of my product to that power.
That means I'd have 9 squared
times s squared
to the 2nd power,
and then I need to raise
t to the 5th to the 2nd power.
Okay?
I've handled the numerator.
Let's go to the denominator.
I need to raise each individual
piece to the 2nd power.
So, 4 squared--
handled that.
I have p to the 3rd squared.
Then I have q
to the 8th squared.
I'll write q the same way
so you don't think that's a 9.
Let's just
clean this up.
I know that 9 squared is 81,
so I've handled that.
I apply my rule about
raising a power to a power.
Multiply those
exponents together--
2 times 2, that's 4.
So, s to the 4th.
Apply that same rule here,
right?
So, t to the 5th raised
to the 2nd power.
5 times 2, that's 10.
So t to the 10th.
I've handled the numerator.
Now on to the denominator.
I know 4 squared,
that's 16, right?
Now I have p cubed
raised to the 2nd power.
So multiply those
exponents together.
3 times 2, that's 6.
So p to the 6th.
Then I have q to the 8th
raised to the 2nd power.
Apply the same rule--
8 times 2, that's 16.
So q to the 16th.
All right?
You're all done.
Got it all simplified, okay?
Getting a little
more comfortable?
Let's try another one.
OK, 2c to the 5th,
d to the 6th,
f to the 4th over 3g
to the 7th, t squared--
We raise all that
to the 3rd power.
It is a mouthful.
Let's get going, though.
I'm going to split it up.
I'm going to raise my
numerator to the 3rd power.
So, 2c to the 5th,
d to the 6th, f to the 4th.
I'm going to cube that.
Then same for my denominator.
3g to the 7th,
t squared, and I'm going
to cube that, right?
I'm going to come down
to get more room.
I'm going to raise
each individual piece
of my numerator
to the 3rd power.
So I have 2 to the 3rd
times c to the 5th
to the 3rd.
Then I've got to handle
that d to the 6th,
so times d to the 6th
to the 3rd,
times f to the 4th
to the 3rd.
Okay?
Then for my denominator,
I have 3 cubed,
All right?
And I'm going
to multiply that
times g to the 7th
to the 3rd power.
Okay, then I'm going
to multiply that
times t squared
to the 3rd power.
Okay? All right.
Now that I have it
all broken up,
I need to handle
simplifying this.
Let's find some little more
room for us to work with.
Get rid of this because
you've handled this part.
I'm going to get it
out of our way
and I'm going to bring up
what we just did.
All right, now we got a little
more room to work here.
Let's clean up
that eraser.
Let's bring up to our 2;
it disappeared.
All right, here we go.
So now, 2 cubed.
I know that's 8--
2 times 2 times 2.
Then I have c to the 5th
raised to the 3rd power.
I know I need to multiply
my exponents together.
So 5 times 3,
that's 15.
I have d to the 6th
raised to the 3rd power.
Multiply your
exponents there.
So, 6 times 3,
that's 18.
I have f to the 4th
to the 3rd power.
I multiply my exponents
there as well.
So 4 times 3,
that's 12.
So, f to the 12th,
all righty?
All over--
let's handle this denominator.
3 cubed, we know that's 27--
3 times 3 times 3.
g to the 7th raised
to the 3rd power,
multiply your exponents
together there.
Seven times 3, that's 21.
So, g to the 21st.
Then we have t squared
raised to the 3rd power.
Multiply your exponents here--
2 times 3, that's 6.
So, t to the 6th.
And you're all done.
All right, I believe
it's your turn.
Go ahead and pause me.
Take a few minutes;
work through these problems.
When you're ready
to compare answers, press play.

(female describer)
Title: Simplify.
1. 7h to the 4th, k to the 5th
over 8j to the 9th,
m to the 6th, all squared.
2. 4a to the 10th, b squared,
c to the 7th over 3d
to the 5th,
e squared, all cubed.

(teacher)
Ready to check?
Let's check.
For the first one,
we had 7h to the 4th,
k to the 5th
over 8j to the 9th,
m to the 6th, and we were
squaring that quotient.
You should've got 49h
to the 8th, k to the 10th
over 64j to the 18th,
m to the 12th, okay?
For the second one,
4a to the 10th, b squared,
c to the 7th over 3d
to the 5th, e squared.
We were raising that quotient
to the 3rd power.
Your answer should've been--
there we go--
64a to the 30th,
b to the 6th,
c to the 21st over 27d
to the 15th, e to the 6th.
If you want to see how I did
either of these, here we go.
Let's get the pen back.
All righty.
For this one, we applied our
same rule we've been applying.
I do a little
shifting here
because I'm running
out of space.
There we go.
I broke our quotient up.
I raised the numerator
to the 2nd power.
So, 7h to the 4th,
k to the 5th squared.
And I raised the denominator
to the 2nd power--
8j to the 9th,
m to the 6th squared.
Then I broke up those
products even further.
For the numerator,
I raised 7 to the 2nd power.
Then I needed to raise h
to the 4th to the 2nd power.
Then I needed to raise
k to the 5th to the 2nd power.
Then I'd handle my numerator.
Same process
for the denominator.
I need to raise 8
to the 2nd power.
Then I need to raise
j to the 9th to the 2nd power.
Then I needed to raise
m to the 6th to the 2nd power.
All righty.
Then we simplify this.
So, 7 squared, that's 49.
We're raising
a power to a power,
so multiply those exponents--
4 times 2 is 8.
So, h to the 8th.
Same thing here--
multiply our exponents
together right there.
So 5 times 2, that's 10.
So, k to the 10th.
For the denominator,
8 squared--
I know that's 64.
We have j to the 9th
raised to the 2nd power,
so multiply those
exponents together.
We've got j
to the 18th.
We have m to the 6th
raised to the 2nd power--
6 times 2 is 12,
so m to the 12th.
All righty?
And that's how we got
that first answer.
Now if you need to see
the second one, here we go.
I followed my same process.
I broke it up to start.
For the numerator,
I had 4a to the 10th, b squared,
c to the 7th, and I raised that
to the 3rd, that whole product.
Then for the denominator,
I had 3d to the 5th, e squared,
and I raised that whole product
to the 3rd power, right?
Then we just broke up
those individual products.
And I will come down here.
So I have 4 to raise to
the 3rd power in the numerator,
times a to the 10th
raised to the 3rd power...
times b squared
raised to the 3rd power...
times c to the 7th
raised to the 3rd power.
All righty.
We handled
the numerator.
Now for the denominator,
we had 3 to the 3rd.
We have d to the 5th
that we raise
to the 3rd power...
And we have e squared
that we need to raise
to the 3rd power.
This was one
of those long ones.
Let's make more room here.
Let's get rid of that.
We've taken care of that.
Let's bring
our current work up.
We made a little more room
to work here.
All righty.
Now 4 cubed,
I think we did that earlier--
4 times 4 times 4, that's 64.
We've got that first bit,
and then your
power-to-a-power rule.
Multiply your exponents there--
10 times 3, that's 30.
So a to the 30th.
Keep applying that rule--
2 times 3 for those exponents,
that's 6.
So b to the 6th.
And for c--
7 times 3, that's 21.
So c to the 21st.
Okay? And the same
process for our denominator.
3 to the 3rd, that's 27.
Power-to-a-power rule
right here--
5 times 3, that's 15.
So, d to the 15th.
Then here we have e squared
to the 3rd power.
Multiply
those exponents there--
2 times 3 is 6.
So, e to the 6th.
That is how I got
the second one.
I hope that you're feeling more
comfortable with these problems
and that your
knowledge of exponents
took you through this.
See you soon.
Bye, guys.