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# Thinking more about negative exponents

Video transcript

In the last video,
we gave an argument for why 3 to the
negative 2 power should be equivalent to 1 over
3 to the positive 2 power. I now want to do a
very similar argument, but I want to do it for general
bases and general exponents. So in particular, I
want to think about what a to the b power,
or in particular, what a to the negative b
power should be equal to. So we know that we want
to uphold the property that, if we have the base
raised to an exponent and then that's being multiplied
by that same base raised to another exponent,
that we would just add the two exponents. So just to be clear, we want
to maintain this property, that a to the b power times
a to the negative b power should be equal to a to
the b plus negative b power, which of
course is going to be equal to a to the 0-th power. And as long as a
does not equal to 0, this should be equal to 1. Let me write that--
for a does not equal 0. So in general-- actually I'll
just copy and paste this-- a to the b times a to
the negative b-- copy and paste-- should
be equal to 1. So this right over here
needs to be equal to 1. And so if we want think about,
what does a to the negative b equal? We just divide both
sides by a to the b. And we would get a
to the negative b. I've kept the colors
consistent this long. Gotta keep going. a to the negative b is
equal to 1 over a to the b. And so I just want
to review this. And it's really
important to keep in mind because when you see
something as a base to a negative exponent,
there's a temptation to somehow introduce
a negative number. But a negative
exponent is really about taking the reciprocal. So you're taking the reciprocal
of this number raised to the positive base. And we've just
shown the argument that if we want this
property be true-- and a lot of the
properties of exponents are all about just keeping
consistency, keeping the old properties continuing
to be true as we introduce new definitions and
new properties-- and we see that if we
want this to be true, then a to the negative b
should be 1 over a to the b. And in other videos we'll
show other motivations for why a negative exponent is
essentially defined this way. But now that we've
seen this is true, for any non-zero
a and any b, let's actually do some examples. And actually, let's think about
the size of these numbers. So for example-- and let
me introduce new colors right over here. I said I would
introduce new colors, and I didn't-- if I were to
take 5 to the third power, we know that's 5
times 5 times 5. That is equal to 125. It's a reasonably large number. Now what should 5 to
the negative 3 power be? And I encourage you
to pause this video and think about this
before I tell you. Use what I just told you. What should 5 to the
negative 3 power be? Well, I assume you've
paused the video. So this is going to
be equal to 1 over 5 to the positive 3 power,
which is equal to 1/125. So this is really interesting. 5 to the third was a
reasonably large number. While 5 to the negative 3,
it wasn't a negative number. It's still a positive number,
but it's a very small number. It's 1/125. Let's do two more examples. Let's say I were to
take negative 1/2, and I were to raise
this to the third power. What is that going to be? Well, it's negative 1/2
times negative 1/2 times negative 1/2, which
is, of course, going to be equal
to negative 1/8. Now what should negative
1/2 to the negative 3 power be, based on everything
that we just talked about? And once again, I encourage
you to pause this video and think about it. Well it should be equal
to 1 over negative 1/2 to the positive 3 power. Well, we just figured
out what that is. Negative 1/2 to the positive
3 power is negative 1/8. This thing right over
here should be equal to negative 1/8. So it's going to be equal
to 1 over negative 1/8. And what's that equal to? Well that's equal to
1 times the reciprocal of this denominator over here. So it's 1 times
negative 8/1, which is going to be
equal to negative 8. So this is interesting. Normally when you take a
number whose absolute value is less than 1 and you take it to
larger and larger exponents, it gets smaller and
smaller and smaller. When you multiply either a
positive or negative 1/2 times 1/2 times 1/2, its magnitude,
its absolute value, is normally going to get
smaller and smaller and smaller. But now, when you raise
it to a negative exponent, it's absolute value is getting
larger and larger and larger. And that's because, once again,
you are taking the reciprocal. The important thing I
want to emphasize here is that you shouldn't
view this negative as when you think about
negative multiplication. Notice, this negative
did not change the sign of what the
eventual value is. You've got negative 1/8. You got negative 8. What it did is, it
changed the magnitude. So this is negative 1/8. When you took to the
negative third power, you took the reciprocal of
that, which is negative 8.