- If we think about something like two to the third power,
we could view this as taking three twos
and multiplying them together, so two times two times two,
or equivalently, we could say this is the same thing
as taking a one and then multiplying it by two three times,
so actually, let's just go with this definition
right over here, and this, of course,
is going to be equal to eight.
Now what would, based on this definition I just did,
what would two to the second power be?
Well, this would be one times two twice,
so one times two times two, which,
of course, would be equal to four.
What would two to the first power be?
Well, that would be one, and we would
multiply it by one two, one times two,
which, of course, is equal to two.
Now let's ask ourself an interesting question.
Based on this definition of what an exponent is,
what would two to the zeroth power be?
I encourage you to just think about that a little bit.
If you were the mathematics community,
how would you define two to the zero power
so it is consistent with everything that we just saw.
Well, the way we just talked about it, we just said
exponentiation is you start with a one and you multiply it
by the base zero times, so we're not gonna multiply it
by any two, so we're just gonna be left with a one.
So does this make sense that two
to the zero power is equal to one?
Well, let's think about it another way,
and let's do a different base.
That was with two, but let's say we have three
and I could say three to the fourth power, that's
three times three times three times three
which is going to be equal to 81,
and let me just write down that
this is going to be equal to 81.
If I said three to the third power,
that's three times three times three which is 27.
Three to the second power is equal to nine.
Three to the first power is equal to three.
Do you notice a pattern every time we
decrease the exponent here by one?
We want three to the fourth, and now we
go three to the third.
What happened to the value?
Well, going from 81 to 27, we divided by three, and
that makes sense, because we're multiplying
by one less three, so we divide by three
to go from 81 to 27, we divide by three again
if our exponent goes down by one,
and we divide by three again when we go from nine to three,
divide by three, so based on this, what do you think
three to the zero power should be?
Well, the pattern is every time we
decrease our exponent by one, we divide by the base,
and so we should divide by three again
would be the logic if we follow that pattern,
and so three divided by three would get us one again.
So I know it might seem a little bit counter intuitive that
something to the zeroth power is going to be equal to one,
but this is how the mathematics community has defined it
because it actually makes a lot of sense.
Either if you view an exponent as taking a one
and multiplying it by the base the exponent number of times,
so I'm gonna multiply one by two three times,
or if you just follow this pattern, every time
you decrease the exponent by one, you're
going to be dividing by the base.
Either of those would get you to the conclusion that
two to the zero power is one, or three to the zero power
is one, or frankly, any number to the zero power is one.
So if I have any number, let's say I have some number a
to the zero power, this is going to be equal to one.
Now I have an interesting question for you,
and let's just say this is the case
when a does not equal zero.
I'll leave you a little bit of a
puzzle for you to think about.
What do you think is zero to the zeroth power?
What should zero to the zero power be?
And what's interesting about zero to the zero power
is you'll get a different answer if you use this technique
versus if you use this technique right over here.
This technique would actually get you to being one,
while this technique would have you divide by zero,
which we don't know how to do.
Anyway, I'll leave you there to ponder
the mysteries of zero to the zeroth power.