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

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to a negative exponent
is equivalent to 1

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over that base raised to
the positive exponent,

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I should say.

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And to get more intuition
about why this makes sense,

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I look at different
powers of 2, and then

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

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integer exponents below 0.

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

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Well, 2 to the third
power is 2 times

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

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Now what about 2 to
the second power?

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

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is of course equal to 4.

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And to go from 2
to the third to 2

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to the second power,
what happened here?

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Well, we divided by 2.

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Now, let's keep going.

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What about 2 to the first power?

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

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

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Now things are going
to get interesting.

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2 to the 0-th power,
and actually this

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

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

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

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

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

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And this is part
of the motivation

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

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But let's keep going.

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What should 2 to
the negative 1 power

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be, if we want to be
consistent about continuing

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to divide by 2?

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Well, we divide by
2 again, and so this

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is going to be equal to 1/2.

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

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2 to the 1 is equal to 1.

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This is equal to the
reciprocal of this.

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Let's keep going.

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This is fun.

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So what should 2 to the
negative 2 power be?

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Well, we should
divide by 2 again.

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

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I think you see the pattern.

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

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

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to the third power.

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So once again,
another way to think

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about why negative exponents
are about taking reciprocals.

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Taking something to the negative
exponent is equivalent to 1

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over taking that same base
to the positive exponent.