Hey, guys.
Welcome to Algebra 1.
Today we'll focus on
solving problems
involving finding
power of a power.
What you know about
exponents and multiplying
and patterns
will go a long way
with these problems.
Ready to get started?
Let's go.
Okay,
to get going with these,
I first want to establish
a pattern for you.
It'll help these type
of problems make more sense.
Knowing what you do
about exponents,
you know that 4
to the 3rd power
is the same as 4 times 4
times 4,
if you expanded it out
to be a product
of its factors.
I have a product
of three 4s, okay?
Applying that knowledge,
how about this one?
What if I needed to raise
4 squared to the 3rd power?
Let's still think about it.
Whatever's in our parentheses,
that term, we're going to
multiply by itself
three times.
So, 4 squared raised
to the 3rd power
is the same as 4 squared
times 4 squared
times 4 squared.
I'm just multiplying 4 squared
times itself three times.
Let's keep breaking this down
and see what it simplifies to
even more.
I know that 4 squared means
that I'm multiplying
4 by itself right?
That first little bit there,
that's the same
as 4 times 4.
4 squared is the same
as 4 times 4.
The middle one
is also 4 squared.
That's another 4 times 4.
Same thing
for the last one.
That's another 4 squared.
That's another
4 times 4, right?
With me so far?
I took that
from 4 squared
times 4 squared
times 4 squared,
and broke it down
even further;
4 squared is the same
as 4 to the 4th,
same here, same here.
Let's get rid of
those parentheses
so we can just see
exactly what's going on here.
I'm just going to rewrite it
without the parentheses.
Break those 4s
out of the groups.
When I see this
expanded out,
how many 4s
am I multiplying by itself?
Let's see--I've got one, two,
three, four, five, six.
That means I can represent this
as 4 to the 6th.
When I expand it all out,
I started at 4 squared
raised to the 3rd power.
When broken down,
it's the same as 4 to the 6th.
Let's keep applying that
and get through
a couple more problems.
I'll do a few of them.
You'll notice the pattern
as I get to a certain point.
Same thing, you know
what we do about exponents.
If I'm raising 3 to the 4th
to the 2nd power,
that means I'm multiplying
3 to the 4th by itself.
I have 3 to the 4th
times 3 to the 4th.
I'm going to expand this out
and really look at it
as the product of 3s.
3 to the 4th,
that means I'm multiplying
3 by itself four times.
That first bit,
that's the same as 3 times 3,
times 3 times 3.
3 to the 4th,
product of four 3s.
The second bit
is also 3 to the 4th,
so 3 times 3
times 3 times 3.
I just expanded that out.
Now I'll scrap
those parentheses
and look at these 3s
and see how many
I've got.
I've got one,
two, three, four here,
multiplied by another one,
two, three, four over here.
How many 3s am I multiplying
together here?
Let's count them out.
One, two, three, four, five,
six, seven, eight.
That means I could
represent this as 3 to the 8th.
So, 3 to the 4th,
raised to the 2nd power,
is the same thing
as 3 to the 8th.
Let's keep going.
Have you noticed
the pattern yet?
If not, don't worry about it,
you will.
This one, I've got X squared,
raised to the 5th power.
So I'm multiplying X squared
by itself five times.
So X squared times X squared
times X squared
times X squared
times X squared.
I bet you're realizing why
someone sought a pattern,
because depending on how large
those exponents are,
you could be writing all day.
Let's keep going
with this.
Each of those X squared
means the same thing
as X times X.
You're just multiplying
X by itself.
For that first X squared,
you've got X times X.
Same for the second one.
Same for that third one,
another X times X.
Same for that fourth one,
another X times X.
Same for that fifth one--ooh!
Didn't write it,
just copied it.
Let's erase.
There we go, bring it back.
I'll break that down.
X times X.
And that last one,
X times X.
Okay, so we wrote it
in groups like this.
Now let's break all those Xs
out of the parentheses
and see what
we're working with.
I've got X times X
times X
times X times X times X
times X times X
times X times X.
How many Xs are you
multiplying together?
Let's count them out.
One, two, three, four, five,
six, seven, eight, nine, ten.
You multiplied X by itself
ten times.
That means you could
represent that product
as X to the 10th.
So X squared
raised to the 5th power
is the same thing
as X to the 10th.
Keep with me.
Let's look at
those three problems
and their answers
all together.
Looking at them
all together,
you may notice the pattern.
When we raised 4 squared
to the 3rd power
a few problems ago,
that answer was 4 to the 6th.
When we raised 3 to the 4th,
to the 2nd power,
that answer was 3 to the 8th.
And when we raised X squared
to the 5th power,
that answer was X to the 10th.
Pause me for a minute.
Look at that.
See if you can figure out
the pattern.
Do you notice any relationship
between the exponents
in my problem,
and then the exponent
in the answer
for each of those?
Pause me
and think about that.
Did you see it?
Did you figure it out?
This is what was going on.
If you multiply these
exponents together, 2 times 3,
that's 6.
If you multiply these together,
4 times 2, that's 8.
If you multiply these together,
2 times 5, that's 10.
That's the pattern with these.
Any time you raise
a power to another power,
the shortcut is to
multiply the exponents together.
Don't go through
all that work like we did--
expanding it all out,
finding the product,
looking at all those factors.
You can just follow
the rule
for raising a power
to a power.
The rule is just this:
whatever your power is
on the inside,
whatever you're raising to,
you just multiply
those exponents together.
Just to say it generally,
you'll see it like this,
maybe with different letters,
but A to the R,
raised to the S power--
it's the same
as A times R to the S.
Multiply
your exponents together
and you'll get your problem
all simplified.
Let's try a couple.
We're just going
to use the rule.
Don't expand it;
use that rule
of raising a power
to another power.
Here, I've got 5 cubed,
raised to the 10th power.
All I need to do here
is just multiply my exponents.
Just multiply the exponents.
That is the shortcut here.
So 3 times 10, that's 30.
That means that
this simplifies to
5 to the 30th.
If you took the time
to expand that,
and write all those fives,
you'd be multiplying
30 fives together.
Nobody wants to multiply
30 fives together,
so represent it like this:
5 to the 30th.
Let's try a couple more.
I've got P to the 3rd
raised to the 11th power.
Remember, just multiply
your exponents together.
I'll come off to the side.
3 times 11, that's 33.
That means that this simplifies
to P to the 33rd power.
You're all done; that would be
your answer right there.
P cubed raised to the 11th power
is the same as P to the 33rd.
All right, let's keep going.
All right, so your turn.
Look at these,
go ahead and pause me.
See how you do
with these.
To compare your answers,
press play
and we'll see how we did.

(female narrator)
Simplify the following
expressions.
Number 1: 3 to the 9th,
all to the 2nd.
Number 2: W to the 7th,
all to the 5th.
Okay, ready to check?
Let's see.
3 to the 9th,
raised to the 2nd power,
that is the same
as 3 to the 18th.
And W to the 7th,
raised to the 5th power,
that's the same
as W to the 35th.
To see how I did those,
in case you got
something different,
here's what I did.
So 3 to the 9th, squared,
I'm raising a power
to another power.
So just multiply
my exponents together.
9 times 2, that's 18.
That means that this simplifies
to 3 to the 18th.
All done.
That next one,
I had W to the 7th,
raised to the 5th power.
So just multiply
those exponents together.
7 times 5, that's 35.
That means this simplifies
to W to the 35th,
and you're all done.
All right?
All right, guys.
Hope you've mastered
solving problems
involving having to raise
a power to another power.
Apply your rule
of multiplying
your exponents together,
and I'll see you soon
for some more Algebra 1.
Bye!