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# Negative exponents

Learn how to rewrite expressions with negative exponents as fractions with positive exponents. Created by Sal Khan.

Video transcript

We already know that
2 to the fourth power can be viewed as
starting with a 1 and then multiplying
it by 2 four times. So let me do that. So times 2, times
2, times 2, times 2. And that will give us, let's
see, 2 times 2 is 4, 8, 16. So that will give us 16. Now I will ask you a more
interesting question. What do you think 2 to
the negative 4 power is? And I encourage you to pause
the video and think about that. Well, you might be
tempted to say, oh maybe it's negative 16 or
something like that, but remember what the exponent
operation is trying to do. One way of viewing it
is this is telling us how many times are we going to
multiply 2 times negative 1? But here we're going to
multiply negative 4 times. Well, what does negative
traditionally mean? Negative traditionally
means the opposite. So here this is how many times
you're going to multiply. Maybe when we make it
negative this says, how many times are
we going to, starting with the 1, how many times
are we going to divide by 2? So let's think about
that a little bit. So this could be
viewed as 1 times, and we're going to
divide by 2 four times. Well, dividing by 2 is the same
thing as multiplying by 1/2. So we could say that
this is 1 times 1/2, times-- let me just
do it in one color. So 1 times 1/2, times
1/2, times 1/2, times 1/2. Notice multiplying
by 1/2 four times is the exact same thing as
dividing by 2 four times. And in this situation
this would get you, well 1/2, well 1 times 1/2 half
is just 1/2, times 1/2 is 1/4, times 1/2 is 1/8,
times 1/2 is 1/16. And so you probably see
the relationship here. If you're-- this is essentially
you're starting with the 1 and you're dividing
by 2 four times. You could also say that 2-- I'm
going to do the same colors-- 2 to the negative 4 is
the same thing as 1/2 to the fourth power. Let me color code
it nicely so you realize what the
negative is doing. So this negative
right over here-- let me do that in
a better color, I'll do it in magenta,
something that jumps out. So this negative
right over here, this is what's causing
us to go one over. So 2 to the negative
4 is the same thing, based on the way we've
defined it just up right here, as reciprocal of 2 to
the fourth, or 1 over 2 to the fourth. And so you could view this as
being 1/2 times 2 times 2 times 2, if you just view
2 to the fourth as taking four 2's
and multiplying them. Or if you use this
idea right over here, you could view it
as starting with a 1 and multiplying it
by 2 four times. Either way, you are
going to get 1/16. So let's do a few
more examples of this just so that we make sure
things are clear to us. So let's try 3 to the
negative third power. So remember, whenever
you see that negative, what my brain
always does is say I need to take the
reciprocal here. So this is going
to be equal to, I'm going to highlight
the negative again, this is going to be 1
over 3 to the third power. Which would be equal to 1/3
times 3 times 3, or 1 times 3 times 3 times 3,
is going to be 27. So this is going to be 1/27. Let's try another example,
I'll do two or three more. So let's take a negative number
to a negative exponent, just to see if we can
confuse ourselves. So let's take the
number negative 4, and let's take it-- I don't want
my numbers to get too big too fast. So let's just take
negative 2 and let's take it to the negative 3 power. I'll make my negatives in
magenta, negative 3 power. So at first this
might be daunting, do the negatives cancel? And that will just be the
remnants in your brain that are trying to think
of multiplying negatives. Do not apply that here. Remember, you see a
negative exponent, that just means the reciprocal
of the positive exponent. So 1 over negative 2
to the third power, to the positive third power. And this is equal to 1
over negative 2 times negative 2 times negative 2. Or you could view it as
1 times negative 2 times negative 2 times
negative 2, which is going to give you 1 over
negative 8 or negative 1/8. Let me scroll over
a little bit, I don't want to have to
start squishing things. So this is equal
to negative 1/8. Let's do one more example,
just in an attempt to confuse ourselves. Let's take 5/8 and raise
this to the negative 2 power. So once again, this negative,
oh I got at a fraction is a negative here. Remember this just means 1
over 5/8 to the second power. So this is just going
to be the same thing as 1 over 5/8 squared,
which is going to be the same thing-- so this
is going to be equal to-- I'm trying to color code it, 1 over
5/8 times 5/8, which is 25/64. 1 over 25/64 is just
going to be 64/25. So another way to
think about it is, you're going to take
the reciprocal of this and raise it to the
positive exponent. So another way you could
have thought about this is 5/8 to the negative 2 power. Let me just take the
reciprocal of this, 8/5 and raise it to the
positive 2 power. So all of these
statements are equivalent. And that would have
applied even when you're dealing with
non-fractions as your base right over here. So 2, you could say well this
is going to be the same thing. 2 to the negative
4 is going to be the same thing as
taking my reciprocal. So this is going to
be the same thing as taking the reciprocal
of 2, which is 1/2 and raising it to
the positive 4 power.