Hi, guys.
Welcome to Algebra 1.
Today we'll focus
on the quotient of powers.
You'll use what you know
about exponents, division,
patterns to help you work
through these problems.
Ready to get going?
Let's go.
Okay, to help you
understand these,
I'll take a little time
going back
to reducing fractions.
Now, probably sometime
in elementary school,
you learned to
reduce a fraction
by finding
the greatest common factor
between the numerator
and the denominator,
then starting
to cancel things out.
So, you have
a 15 in the numerator,
a 20
in the denominator.
Think about
the factors of 20 and 15.
Factors are the numbers
that you multiply together
to get your product.
You may have different factors,
like multiplication.
For example, 20.
I could get 20
by multiplying 1 times 20,
10 times 2, 5 times 4.
But I want to consider
the factors--
the one
it has in common with 15.
For example,
I know I can get 15
by multiplying
5 times 3.
I know 5 times 3 is 15,
and I know that I can get 20
by multiplying 5 times 4.
I picked those factors because
they had the 5 in common--
the factors of 20
and of 15.
You have
a 5 in the numerator
and a 5 in the denominator;
you can cancel that out.
Say, "Okay, then 15/20ths
reduces to 3/4ths," right?
You could
just reduce that fraction
without using a calculator.
Keep that in mind
as we work our way
through these problems.
Stay with me.
You might be feeling
a little "ugh"
about these fractions,
but we're going to work
our way through it.
So I see
I have 32 over 28.
Think about the factors
32 and 28 have in common.
I want
the greatest common factor.
Knowing what you know
about factors,
I know that I could
multiply 8 times 4,
and that would give me 32.
And I know that I could
multiply 7 times 4,
and get 28.
So the greatest common factor
between those
two numbers is 4.
Cancel out the 4s,
and you could say,
"Okay, then that fraction
simplifies to 8/7ths."
Right?
Just reduce that fraction.
Keep that in mind.
Let's keep going.
So there's
some exponents in here.
You have 2 to the 5th
divided by 2 squared.
Let's just expand those.
Expand that numerator,
expand that denominator.
Look at it
as a product of its factors,
of a bunch of 2s.
The numerator
is 2 to the 5th,
so that's the product
of five 2s:
2 times 2 times 2
times 2 times 2, all right?
In my denominator,
I have 2 squared,
so that's the product
of two 2s, all right?
Okay, so 2 times 2.
Keep in mind what we just did
with those fractions.
Cancel out factors
that they have in common.
Cancel out a 2
in the numerator
with a 2 in the denominator--
that would cancel out.
Cancel out another 2
in the numerator
with this 2
in the denominator,
and that would just leave me
with those three 2s up top.
So, 2 times 2 times 2.
If I wanted to simplify that,
I could represent it
as 2 to the 3rd power.
So 2 to the 5th
divided by 2 squared
simplifies to 2 to the 3rd,
okay?
Keep doing
a few more of these with me.
You're going to notice
the pattern.
I've got 8 to the 6th
divided by 8 to the 4th.
That means in my numerator
I have 8 multiplied by itself
six times, okay?
So 8 times eight times 8
times 8 times 8
times 8.
Let's double check that.
One, two, three,
four, five--yep, six 8s.
Then in my denominator,
I have 8 to the 4th,
so that's the product
of four 8s.
So 8 times 8 times 8
times 8, okay?
I've expanded the numerator
and denominator
to look at it
as a product of 8s.
What can we cancel?
You remember one 8
in the numerator
will cancel out one 8
in the denominator.
So I can cancel out
that pair.
That's another pair.
That's
another pair.
That's another pair,
all right?
I ran out of 8s
in the denominator,
with two left
in the numerator.
8 times 8,
that's 8 squared.
If you look
at where we started,
8 to the 6th
divided by 8 to the 4th
is 8 squared, okay?
Let's keep going.
Are you picking up
on the pattern here?
I've got X to the 7th
divided by X to the 3rd.
What do we have on top?
The product of seven Xs.
So X times X
times X times X
times X times X times X.
Double check that--
make sure we got seven.
One, two, three,
four, five, six, seven.
So, we have X
to the 7th on top.
Denominator,
we have X cubed.
So that's X times X
times X, okay?
What do we do now?
See what you can cancel out.
An X in the numerator
will cancel out with one
in the denominator.
Another pair cancels out,
and another pair
can cancel out.
When I do that,
I'm left with four Xs
in my numerator,
the product of four Xs.
So X times X
times X times X.
That's X to the 4th power.
So X to the 7th
divided by X cubed
equals X to the 4th.
Okay?
Let's keep going.
Let's look at
these three all together.
When put together--
you may notice
the pattern happening here.
When we had 2 to the 5th
divided by 2 squared,
that answer was 2 to the 3rd.
8 to the 6th
divided by 8 to the 4th,
that was 8 squared.
X to the 7th
divided by X cubed,
that was X to the 4th,
okay?
Take a minute.
Pause me for a second.
Look at these three problems,
what we started with
and what we ended with.
Pay close attention
to the exponents.
Do you notice
a relationship?
Something's going on
with the exponents
that we start with and then
the one that we end with.
Did you notice it?
Okay, I've got my pen.
5 minus 2, that's 3.
6 minus 4, that's 2.
And 7 minus 3, that's 4.
Taking the quotient
of powers, the shortcut,
because we've noticed
this pattern,
is just subtract the exponents
that you start with.
That's how we come upon
this rule
for the quotient of powers.
For two terms
with the same base--
here, it's a--
and you're finding
the quotient,
just subtract their exponents
to get the answer.
You won't have to write
all those factors out,
and cancel out.
Just use this rule
we discovered
from using those problems
to find a pattern.
Let's start practicing
with this rule.
Let's get the pen back.
X to the 18th
divided by X to the 10th.
I have a quotient of powers.
All I need to do here--
my bases are both Xs--
is subtract the exponents.
Here, 18 and 10
are my exponents,
so 18 minus 10,
that's 8.
Notice it's the top exponent,
subtract the bottom one,
okay?
So X to the 18th
divided by X to the 10th,
that's X to the 8th,
and that's your answer.
To apply that rule,
subtract those exponents.
Let's keep going.
Y to the 9th
divided by Y to the 4th,
so same thing.
Apply the rule, just subtract
your exponents.
Come off to the side.
9 minus 4, that's 5.
This would simplify
to Y to the 5th.
If you show the product
of those nine Ys and
of those four Ys,
and canceled out,
you would end up
with the product of five Ys.
The answer is Y to the 5th.
Okay? Let's keep moving.
This one is a little different,
but not too different.
C to the 11th
times D to the 15th
divided by C to the 10th
times D to the 8th.
You have two different bases--
C and D.
Just take them separately,
like grouping
your like terms together.
For example,
I would first handle
that C to the 11th
over C to the 10th.
Apply my rule,
and just subtract my exponents.
So 11 minus 10, that's 1.
That would simplify
to C to the 1st.
With an exponent that's 1,
you don't have to write it.
But I will just to reinforce
that we just subtracted
our exponents.
So, you handled the Cs,
then move on
to the Ds.
You have D to the 15th
over D to the 8th.
Like we just did,
apply your rule,
and subtract your exponents,
right?
So 15 minus 8,
that's 7.
So, this would be
D to the 7th.
When I simplify
those Ds,
I get D to the 7th.
So, when you have
an exponent that's 1,
you generally
never see it written.
You would probably see this
as C, D to the 7th.
If you did write that 1,
you're not incorrect.
This simpler way to write it
is how you'll see it
most often.
Okay? Let's keep going.
This one, you see,
we've got three variables.
G to the 15th, H to the 22nd,
K to the 9th
over G to the 10th,
H to the 18th,
K to the 6th.
We apply our same process
and take
the variables together,
the like ones
at the same time.
Let's deal with the Gs first.
So G to the 15th
divided by G to the 10th.
We're just going to subtract
our exponents.
So 15 minus 10, that's 5.
I know when I handle
the Gs,
I end up
with G to the 5th.
Then, I'd move on
to the Hs.
I have H to the 22nd
over H to the 18th.
Just subtract my exponents.
All right, 22 minus 18,
that's 4.
This is just H to the 4th
when I simplify that.
Last, but not least,
let's get the Ks.
We have K to the 9th
over K to the 6th.
Just subtract my exponents.
9 minus 6, that's 3.
That part would just break down
to K to the 3rd.
My last part of this,
K to the 3rd.
You're all done,
handled all three pieces.

(female narrator)
G to the 5th,
H to the 4th, K to the 3rd.
Okay? Not too bad, right?
All right,
let's keep going.
It is now your turn, okay?
Press pause,
take a few minutes.
Work your way through
these problems.
When you're ready to check
your answers, press play again.

(female narrator)
Simplify the following
expressions.
Number 1: Z to the 30th
over Z to the 20th.
Number 2: M to the 7th
times N to the 8th
over M to the third
times N to the 5th.
Number 3:
X to the 18th times Y
to the 13th times P to
the 4th over X to the 12th
times Y to the 9th
times P squared.
All right, you ready?
Let's see how you did.
C to the 30th
divided by Z to the 20th,
that's Z to the 10th
for that one.
Two: M to the 7th,
N to the 8th over M to the 3rd,
N to the 5th,
was M to the 4th,
N to the 3rd.
All right?
Finally:
X to the 18th, Y to the 13th,
P to the 4th
divided by X to the 12th,
Y to the 9th, P squared.
You should
have got X to the 6th,
Y to the 4th,
P to the second power.
To see how I did these,
keep with me.
How did I get
that first one?
Okay, just applied my rule,
and subtracted my exponents.
So 30 minus 20, that's 10.
That means that this will
simplify to Z to the 10th.
On the next one,
I had two different variables,
but I applied
the exact same process.
I handled the Ms first.
M to the 7th
divided by M to the 3rd,
you just subtract
your exponents.
So 7 minus 3, that's 4.
That means the M piece
of that simplifies
to M to the 4th.
Got the first bit.
All right,
then I have to handle the Ns.
I have N to the 8th
divided by N to the 5th.
Just apply your rule.
Subtract the exponents.
Okay, 8 minus 5,
that's 3,
so N to the 3rd.
Then you've got
the last part of that.
That's how I got number 2.
If you need to see
how I did the last one...
This is the one
with three different variables.
Take them
one at a time--
the same type
at a time.
So X to the 18th
over X to the 12th.
Just subtract your exponents.
18 minus 12, that's 6,
so X to the 6th.
That was the first part.
Then handle the Ys.
So Y to the 13th
over Y to the 9th.
Subtract your exponents,
so 13 minus 9.
That's 4, right?
That's where the Y
to the 4th came from.
Then we have the Ps.
So P to the 4th
divided by P squared.
Subtract your exponents.
4 minus 2, that's 2,
so P to the 2nd power.
That's where the last piece
came from.
That's how I got
those three answers.
There's another kind of problem
that applies this rule.
I want you to know
how to do this type, too.

(female narrator)
Example 5: 40G to the 5th,
Y to the 9th
over 20G squared, Y.
This is still a problem
involving the quotient
of powers,
but you have coefficients
out front.
Now you also
have to handle that.
Come off to the side.
It's not
a very different process.
I need to handle that 40
divided by 20 first.
40 divided by 20,
that's 2, right?
That means the first part
of this is 2.
Then I have G to the 5th
divided by G squared.
G to the 5th
divided by G squared.
Subtract my exponents here.
So 5 minus 2, that's 3,
so I've got G cubed.
Then I've got the Ys
to handle here.
I've got Y to the 9th
divided by Y.
Uh-oh! You see, you don't
really see an exponent?
Remember, when you don't see
an exponent,
there's an invisible 1
for your exponent,
so just throw it in there
to help you
get through it.
I've got Y to the 9th
divided by Y to the 1st.
Subtract those exponents.
So 9 minus 1, that's 8,
so Y to the 8th.
Ooh! I wrote 18.
Let's get rid of that.
Y to the 8th.
All right?
All done with that one.
Okay?
Let's try another one.
This one,
I have 12X to the 9th,
Y to the 11th divided by 15,
X to the 6th,
Y to the 7th.
So same process,
handle the coefficients first.
So 12 divided by 15.
All right?
Don't be alarmed.
That won't divide cleanly
like the other one.
My answer is not
a whole number;
it's another fraction.
We're not afraid
of fractions.
We'll reduce
this fraction.
Remember when we were
reducing the fractions?
Do the same thing.
I need a greatest common factor
between 12 and 15.
It helps to know
your multiplication tables.
All right,
I know 4 times 3 is 12,
and I know 5 times 3 is 15.
The greatest common factor
between those two terms,
those two numbers, is 3,
right?
I can cancel out the 3s.
That means this fraction
reduces to 4/5ths.
The first part
of this problem,
the part involving
the whole numbers,
that part equals 4/5ths,
okay?
I'll keep moving along
and apply my quotient of powers
rule to handle the rest.
So X to the 9th
divided by X to the 6th.
You remember what we do?
Subtract the exponents.
So 9 minus 6, that's 3.
So X to the 3rd power.
I know that's the next bit.
Let's handle the Ys.
You've got that Y to the 11th
divided by Y to the 7th.
Apply your quotient
of powers rule.
Subtract your exponents.
So 11 minus 7, that's 4.
So Y to the 4th.
Sometimes, you won't see
the answer written like this
when it's a fraction.
You should know the different
ways this can be written.
You may see it written
like this--
4/5ths times X
to the third, Y to the 4th.
Or sometimes, people like
to separate the numerator
from the denominator
even further,
like really show
that division.
So, you may see this
as 4X to the 3rd,
Y to the 4th, all over 5.
Those each mean
the same thing.
It's whichever way
you want to write it.
They mean the same thing,
they're both acceptable.
Okay? Let's move along.
It is your turn,
so press pause.
Take your time,
work through these problems.
When you're ready to compare
answers with me, press play.

(female narrator)
Simplify the following
expressions.
Number 1: 75A to the 19th,
B to the 18th over 50A
to the 11th, B to the 10th.
Number 2: 30H to the 9th,
V to the 11th,
T to the 4th
over 5H to the 5th,
V to the 3rd, T.
You ready
to see how you did?
Let's compare our answers.
For the first one,
75A to the 19th,
B to the 18th
divided by 50A to the 11th,
B to the 10th.
That was three halves A
to the 8th, B to the 8th,
or maybe you wrote it as--
maybe you wrote it as this:
3A to the 8th,
B to the 8th over 2.
Either one
would have been fine.
All right, the second one,
30H to the 9th,
V to the 11th,
T to the 4th
divided by 5H to the 5th,
V to the 3rd, T.
You should have got
6H to the 4th, V to the 8th,
T to the 3rd. Okay?
See how I did these?
Then stay with me,
I'll work them out.
For the first one,
I did
what we did before.
I focused on those
coefficients first.
I said,
"Okay, 75 divided by 50."
Those won't
divide evenly, okay?
Whenever I see numbers
25, 50, 75, 100,
I always think quarters
to figure out
what my factors are.
I think like, "75,
that's three quarters," right?
The factors of 75,
25 times 3.
All right?
50, I know I could factor that
with 25 times 2, right?
When I do that I see my greatest
common factor is 25.
I cancel out those 25s.
So I got
the three halves,
the 3 over 2.
You notice that's
an improper fraction.
The numerator is larger
than the denominator.
In elementary school,
people really hated that
and wanted
to make it a mixed number.
When you get to Algebra 1,
nobody hates it anymore.
Leave an improper fraction
as an improper fraction,
it's fine.
It's easier
to leave it like that Algebra.
I'll leave that
as three halves, okay?
Now I'll handle the A part.
A to the 19th
divided by A to the 11th.
I subtract my exponents.
So 19 minus 11, that's 8.
So A to the 8th.
And B to the 18th
divided by B to the 10th.
Subtract my exponents here.
18 minus 10, that's 8,
so B to the 8th.
You just remember,
like I said,
sometimes it's written
like this,
where you're forced
to notice everything
that's in the numerator.
That 3A to the 8th,
B to the 8th,
with that 2 on the bottom.
Both mean the same thing.
I like the first way--
that's my habit--
but either one is fine.
It's completely up to you.
If you see that second one,
same exact process:
take the coefficients first,
and then deal with the rest.
So 30 divided by 5, that's 6.
That's where my 6 came from
in my answer.
Then I handled the Hs.
So H to the 9th
divided by H to the 5th.
Use your quotient
of powers rule.
Subtract the exponents.
So 9 minus 5, that's 4.
That's where I got
the H to the 4th.
Then I handled the Vs.
So V to the 11th
over V to the 3rd.
Subtract your exponents.
So 11 minus 3,
that's 8.
That's where the V to the 8th
came from.
Then the last bit:
that T to the 4th divided by T.
Remember how, when you
don't see an exponent,
there's an invisible 1?
So let's subtract 4 minus 1.
That's 3, so T to the 3rd.
That's where the last bit
of that came from.
All right, guys!
I hope you're using
your quotient of powers rule
to solve your problems,
and that using your knowledge
of exponents,
division, and factors--
it comes together
to help you with
these kind of problems.
Hope to see you soon.