More negative exponent intuition

I don't think we can do enough
videos on why raising something

to a negative exponent
is equivalent to 1

over that base raised to
the positive exponent,

I should say.

And to get more intuition
about why this makes sense,

I look at different
powers of 2, and then

think about what makes sense
as we go to exponents below 0,

integer exponents below 0.

So let's start with
2 to the third power.

Well, 2 to the third
power is 2 times

2 times 2, which of
course is equal to 8.

Now what about 2 to
the second power?

Well that's going to
be 2 times 2, which

is of course equal to 4.

And to go from 2
to the third to 2

to the second power,
what happened here?

Well, we divided by 2.

Now, let's keep going.

What about 2 to the first power?

Well, that's just
2, and once again,

to go from 2 squared to 2 to the
first power, we divided by 2.

Now things are going
to get interesting.

2 to the 0-th power,
and actually this

will help to build the
intuition of why it's something

nonzero to the 0-th
power is defined to be 1.

Well, so far, every time we
decremented the exponent by 1,

we essentially divided by 2,
so we should divide by 2 again.

So if we divide by
2 again, we get 1.

And this is part
of the motivation

of why 2 to the 0 power
should be equal to 1.

But let's keep going.

What should 2 to
the negative 1 power

be, if we want to be
consistent about continuing

to divide by 2?

Well, we divide by
2 again, and so this

is going to be equal to 1/2.

Notice, 2 to the
negative 1 is 1/2.

2 to the 1 is equal to 1.

This is equal to the
reciprocal of this.

Let's keep going.

This is fun.

So what should 2 to the
negative 2 power be?

Well, we should
divide by 2 again.

Divide by 2 again,
you get to 1/4.

I think you see the pattern.

2 to the negative 3 power, well
we should divide by 2 again.

And we get to 1/8, which
is the reciprocal of 2

to the third power.

So once again,
another way to think

about why negative exponents
are about taking reciprocals.

Taking something to the negative
exponent is equivalent to 1

over taking that same base
to the positive exponent.