Hey, guys.
Welcome to Algebra I.
Today's lesson focuses
on the product of powers.
You apply what you know
about exponents and patterns
to work through
these problems.
Ready?
All right, let's go.
Okay, to understand
product of powers,
bear with me for a minute
while I take you through
problems to set up patterns.
It'll help you
understand it once we get going.
The meaning of 4 to the 3rd--
What does that
really represent?
My exponent tells me
how many times
I'm multiplying
a number times itself.
So, 4 to 3rd power
means the same thing
as 4 times 4 times 4.
It's the product
of three 4s, right?
Okay.
5 to the 4th.
Same idea, right?
The exponent tells you
how many times
to multiply 5 by itself.
In this case,
because my exponent is 4...
I'm going to multiply 5
times 5 times 5 times 5.
I've got the product
of four 5s.
Okay?
Now, let's keep going.
x to the 5th--
same idea.
Your exponent is 5,
so I should see
the product of 5 "x"s.
So, one, two, three,
four, five.
All right, x to the 5th.
You start seeing what's
going on with these exponents.
What if you have
something like
5 squared times
5 to the 4th?
What does that
really represent?
Let's break it down.
I know that 5 squared--
that's 5 times 5.
Then I'm multiplying that
times 5 to the 4th.
So, I know 5
to the 4th--
I should see the product
of four 5s for that part,
so 5 times 5
times 5 times 5.
When I look at the problem,
I have the product
of one, two, three,
four, five, six 5s.
I could say that 5 squared
times 5 to the 4th
is the same as 5 to the 6th.
When I represent that,
expand it all out,
look at it all together,
I have the product
of six 5s, right?
Keep going.
What about 3 cubed
times 3 squared?
Use that same idea
we did on the last one
and see what happens
with this one.
I know 3 to the 3rd,
that's the product
of three 3s.
3 squared,
that's the product of two 3s.
When I look at this,
I'll expand it like this.
I see that I have the product--
count them out--of five 3s.
So 3 to the 3rd--oop--
let's get that pen back.
Let's get this out of my way.
Let's get that pen back.
There we go.
3 to the 3rd times 3 squared
is the same
as 3 to the 5th.
When you look
at that product expanded,
you have the product
of five 3s, okay?
All right.
Look at this one:
x to 3rd times x to the 4th.
Getting the hang
of what's going on here?
I know x to the 3rd--
that's x times x times x,
the product of 3 "x"s.
I'm multiplying that
times x to the 4th,
so the product of 4 "x"s--
one, two, three, four.
When I look at that product
expanded out,
I have one, two, three,
four, five, six, seven--
I've got
the product of 7 "x"s.
So x to 3rd times x to the 4th
is the same as x to the 7th.
Okay, I wonder if you started
to notice a pattern.
If not,
let's try one more.
We'll bring it together
and see if you can spot it.
For this one,
I've got 4 times 4 squared.
When you don't see
an exponent,
you can pretend there's
an exponent of 1
because that doesn't
change anything.
That's just one little 4
standing on its own.
Then you're multiplying it
times 4 squared.
That is the product
of two 4s.
When I look at that
product expanded,
I see I have
the product of three 4s.
That means that 4
times 4 squared--
remember, we could think
about that as 4 to the 1st--
is the same
as 4 to the 3rd, right?
We proved that
by expanding it all out.
Let's look
at everything and see
if you notice
what's going on.
These are the problems
we did.
We're comparing their answers
all together on one page.
We remember 5 squared
times 5 to the 4th
gave us 5 to the 6th.
3 square--I'm sorry--
3 cubed times 3 squared,
that was 3 to the 5th.
x cubed
times x to the 4th--
that was x to the 7th.
We said to put a 1 there.
Think about it like that--
4 to the 1st times 4 squared--
that's 4 to the 3rd.
Pause the tape if you want,
and look at these answers
and problems.
Look at how they started
and see if you can figure out
a pattern.
Did you notice a relationship
between those exponents?
Take a look back here.
In this problem,
my exponent was 2 and 4.
Now, 2 plus 4 equals 6.
For this one,
my exponent was a 3,
exponent here was a 2.
3 plus 2, that equals 5.
I had exponents of 3
and of 4.
3 plus 4, that's 7.
Exponent of 1, exponent of 2--
my exponent
for my answer was 3.
The pattern
with these kind of problems,
if you're multiplying
two terms
that have the same base,
like that same--
Here it's 5, here it's 3.
The base is the same.
Add your exponents together
to get the answer.
That is how
we came about the rule
which is called
"product of powers."
When multiplying two terms
with the same base
and they have exponents,
to get the answer,
add the exponents.
You don't have to go through
and expand that problem
and write out those factors,
because you know a shortcut--
just use the rule.
Add the exponents together
when multiplying
and your bases
are the same.
Look at this one.
We'll use the rule.
Make sure
I got my pen.
I got y to the 8th
times y to the 4th.
My bases are the same.
They're both y.
I simplify this
by adding my exponents.
I'll come off
to the side.
You're adding
your exponents.
Okay, so 8 plus 4,
that's 12.
That means that y to the 8th
times y to the 4th,
that's just y to the 12th.
You just add your exponents
together, and you're all done.
You simplify that, okay?
Let's keep applying this rule.
Keep going.
7 to the 10th
times 7 to the 15th.
Same exact thing.
We add those
exponents together
because our bases
are the same.
I'll come off to the side--
10 plus 15, that's 25.
That's what I get
when I add those exponents.
That means the answer
is 7 to the 25th,
and you're all done.
That's it.
You simplify that also, okay?
Let's keep moving.
It's your turn now.
Pause the tape.
Take a few minutes.
Work through
these problems,
and we'll get together
and compare our answers.

(female describer)
1. x to the 11th power
times x to the 5th power.
2. 5 to the 8th power
times 5 to the 20th power.

(teacher)
Okay, took some time.
Let's compare our answers.
The first one, x to the 11th
times x to the 5th--
Let's get rid of the pen
so I could reveal this.
That's x to the 16th.
5 to the 8th
times 5 to the 20th.
That's 5 to the 28th.
To see how I did these--
maybe you missed it
or you want
to check your work.
I'll work those out for you.
Get that pen back.
This one, x to the 11th
times x to the 5th,
just need to add
those exponents together--
11 plus 5, that's 16.
That means the answer
to this: x to the 16th.
That's how I got that one--
add those exponents together.
On the second one,
I had 5 to the 8th
times 5 to the 20th.
Just add
your exponents together--
8 plus 20, that's 28.
That means my answer here
is 5 to the 28th,
and you're all done.
Okay.
That's how I did those two.

(female describer)
Example 3: Simplify 6g cubed,
h to the 4th times
2g to the 10th,
h to the 5th.
All right, we're going
to apply the same rule here.
This problem
is a little different.
If you notice, we are still
multiplying two terms together,
but our terms
have a coefficient.
I have a coefficient
of 6 right here
and a coefficient of 2 here.
I'm still just finding
the product of powers.
What I do here,
just to help us understand,
I'm going to rearrange
this multiplication.
The commutative property
tells us
we multiply in any order.
It doesn't change the problem.
I bring
my coefficients out front
so I can look
at those together.
Then I'm going to group
my like exponents--
not the exponents--
but the variables together.
It helps me organize this
and break it down further.
I'm going to put
that 6 times 2 out front.
Get those coefficients
together.
Now I'm going to multiply
g to the 3rd
times g to the 10th
and get those two
terms together,
so I handled those.
I get that h to the 4th
times that h to the 5th
and get those together.
Okay.
Now I look
at everything together.
I keep working my way
through this problem.
Now,
the associative property says
I can group
when I'm multiplying.
It doesn't change anything.
I group those
like terms together
and handle them
in their individual groups.
Right here--
6 times 2, I know that's 12.
For g to the 3rd
times g to the 10th,
I apply the rule,
that product of powers.
I add exponents to get
the answer for that one--
3 plus 10,
that's 13.
I have h to the 4th
times h to the 5th.
Just add those
exponents together:
4 plus 5, that's 9.
So, h to the 9th.
You're all done.
That one is the same idea.
They threw a curve ball
because you got coefficients.
We group
the like things together
and use rules
you already know.
Okay, let's try another one:
8t to the 3rd, j to the 10th
times 5tj to the 7th.
Remember what we just did.
We're going
to rearrange this at first.
I'm going to pull the
coefficients out to the front.
I'll have 8 times 5 out front.
Then I'm going
to take those "t"s,
put them together.
So, t to the 3rd times t.
I'll get those "j"s
and put those together.
So j to the 10th
times j to the 7th.
Okay.
I'm going to use my
parentheses to group them off.
I'm going to handle each group,
each chunk separately.
So, 8 times 5,
I know that's 40.
Then t to 3rd times t--
when you don't
see an exponent,
there's like
an invisible 1 there.
I need to add those
exponents together--
3 plus 1, that's 4.
So, t to the 4th.
I have j to the 10th
times j to the 7th.
Add my exponents together:
10 plus 7, that's 17.
You're all done.
That expression
just simplifies
to 40t to the 4th,
j to the 17th.
Just applying
those rules that we learned.
Now it's your turn.
Try these two problems.
These involve
the coefficients we covered.
Pause the tape,
then play it
when you're ready
to compare answers.

(female describer)
Simplify the following
expressions.
1. 9m to the 6th,
n to the 7th times 5m
to the 4th, n squared.
2. 10x cubed, y to the 5th
times 8x squared,
y to the 6th.

(teacher)
Okay, let's see how you did.
Get the pointer tool.
That first one,
9m to the 6th, n to the 7th,
times 5m to the 4th,
n squared--
should have got 45m
to the 10th, n to the 9th.
The second one, 10x cubed,
y to the 5th times 8x squared,
y to the 6th--
that answer was
80 to the 5th, y to the 11th.
If you need to see how I did
either one, then stay with me.
Let's get that pen
and work this out.
Remember, our process
that we're following.
Bring coefficients out front,
then group
the like terms together.
I'd have 9 times 5
for my coefficients.
Then I have m to the 6th
times m to the 4th.
Then I have n to the 7th
times n squared.
Okay?
Now I throw my parentheses
in there to group them off
and handle them in chunks.
So, 9 times 5, that's 45.
m to the 6th
times m to the 4th--
we're just going to
add those exponents together.
So, 6 plus 4, that's 10.
I have m to the 10th.
Then I have n to the 7th
times n squared.
I'm just adding those
two exponents together--
7 plus 2, that's 9.
That's how
I got that answer:
45m to the 10th,
n to the 9th.
You see the next one?
Keep going.
I have 10x cubed,
y to the 5th
times 8x squared,
y to the 6th,
so same process.
Get those
coefficients together.
10 times 8.
Then I have x cubed
times x squared.
I've got y to the 5th
times y to the 6th.
You remember
what we did next?
Threw those parentheses in,
group them off.
Okay, 10 times 8, that's 80.
I've got x cubed
times x squared.
So just add
those exponents together:
3 plus 2, that's 5.
I've got y to the 5th
times y to the 6th.
Just add those
two exponents together:
5 plus 6,
that's 11, okay?
That's how we got that one.
All right.
Well, you have completed your
lesson on product of powers.
I hope your knowledge
of patterns and exponents
helped you through
these problems.
I hope to see you
here soon. Bye.