Let's review a little bit of what we know about exponents.
So if I were to have 3 to the second power times 3
to the third power, what does that equal?
Well, we could expand it out just as a review.
3 to the second power-- well, that's going to be 3 times 3.
And then 3 to the third power-- we're
going to multiply it by 3 to the third power--
is going to be 3 times 3 times 3.
So what is this going to be equal to?
Well, this is going to be equal to 3 to the 2 plus 3 power.
And this comes straight out of what an exponent is.
We're taking two 3's and multiplying it together.
And then we're going to multiply that with three 3's
being multiplied together.
So you're going to end up with 2 plus 3, or five,
3's being multiplied together.
So this, of course, is going to be equal to 3
to the fifth power.
And that's a little bit of review.
And this is one of the core exponent properties.
If you have the same base multiplied to exponents,
and you're taking the product of those two expressions,
you can essentially, or you do, just add the exponents.
3 to some exponent times 3 to some other exponent
is going to be 3 to the sum of those two exponents.
So that's kind of interesting, but now
let's use this property right over here
to try to think about what a negative exponent should be.
So we're going to try to think about what should-- let's just
say 3 to the negative 2 power-- what should this be equal to?
And I encourage you to pause this video and think about it.
Think about, what should this be equal to if we
want this property to hold, that if you
have the same base raised to different exponents,
and you're taking the product of those two expressions,
that you can take that base to the sum of those two exponents.
Well, let's try a little experiment here.
What should 3 squared times 3 to the negative 2 power be?
What should this be equal to?
Well, if we want to keep this property going,
this should be equal to 3 to the 2 plus
negative 2 power, which, of course, would be equal to 3
to the 0-th power.
And any non-zero number to the 0-th power
is going to be equal to 1.
So this should be equal to 1.
So we should define 3 to the negative 2 in some way
so that 3 squared times 3 to the negative 2
should be equal to 1.
We need to do that if we want this property to hold right
over there.
So let's just think about what 3 to the negative 2
should actually be.
So we want this to equal 1.
Let me copy and paste that so that we can manipulate it
a little bit algebraically.
All right.
So we have that on one side.
We want that to be equal to 1.
And let me clean this up a little bit.
We want that to be equal to 1.
So to define 3 to the negative 2,
or to think about what 3 to the negative 2 should be,
let's divide both sides by 3 squared.
Let's divide both sides by that.
So let me copy that.
So we're going to divide this side by that,
and we're going to divide this side by that.
And let me draw a line so that you know that we're dividing.
So we're going to divide the left hand
by that and the right hand by that.
Well, 3 squared over 3 squared-- those cancel out.
And now we get a really interesting result.
We get that 3 to the negative 2 power
should be equal to 1 over 3 to the second power.
And I really want you to internalize this right over
here.
You have something to the negative power,
to the negative 2 power, in this case.
That's equal to 1 over 3 to the positive version
of that exponent.
And this is true in general.
In the next video, we'll use a very similar argument
to prove this for any non-zero base and any exponent,
that raising it to the negative should
be-- if you want this exponent property to hold,
that you would add the exponents when we're
taking the product of the same base raised
to different exponents.
If you want that property to hold,
when you raise something to the negative exponent,
it should be 1 over that base raised
to the positive version of that exponent.
So 3 to the negative 2 power is 1 over 3
squared, which is equal to 1/9.