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.