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# Zero, negative, and fractional exponents

Video transcript

So a few videos ago, I told
you that anything to the zeroth power is equal to 1. So x to the zeroth power
is equal to 1. And I gave you one argument
why this is the case. I used the example of, if we
have 3 to the first power, that is equal to 3. 3 to the second power
is equal to 9. 3 to the third power
is equal to 27. So every time we decrease by a
power, we're dividing by 3. 27 divided by 3 is 9. 9 divided by 3 is 3. Then 3 divided by 3 is 1. And that should be what 3
to the zeroth power is. So that's one way to
think about it. The other way to think about
it is that we need this for the exponent properties
to work. For example, I told you that a
to the b times a to the c is equal to a to the b plus c. Now, what happens if c is 0? What happens if we have a to
the b times a to the 0? Well, by this property, this
needs to be equal to a to the b plus 0, which is equal
to a to the b. So a to the b times a to the 0
must be equal to a to the b. If you divide both sides of this
times a-- let me rewrite this-- a to the b times a to the
0, if we use this property up here, must be equal to a to
the b, right? b plus 0 is b. If you divide both sides by a
to the b, what do you get? On the left-hand side,
you're left with just a to the 0, right? These cancel out. a to the 0 is equal to 1. And you can use a similar
argument in pretty much all of the exponent properties, that we
need anything to the zeroth power to be equal to 1. And it also makes sense as we
divide by 3, each step as we decrement our exponent. It keeps working. When you take 3 to the negative
1 power, we saw on the last video that that's equal
to 1 over 3 to the first power, or 1/3 . So once again, from 3 to
the 0 to 1/3, you're dividing by 3 again. So it really makes sense on
some level that 3 to the zeroth power is equal to 1. But that leaves a little
bit of a gap. What about 0 to the
zeroth power? This is a very strange notion. 0 multiplied by itself
0 times. And it depends what context
you're using. Sometimes people will say that
this is undefined, but many more times, at least in my
experience, this'll be defined to be 1. And the reason why-- even though
this is completely not intuitive, and you could type
in 0 to the zeroth power in Google, and it'll give you 1. Even though this is completely
not intuitive, the reason why this is defined to be this way
is that makes a lot of formulas work. One in particular, the binomial
formula works for your binomial coefficients,
which I'm not going to go over right here, when 0 to the zeroth
power is equal to 1. So that's an interesting thing
for you to think about, what that might even mean. So let's talk about some of
the other properties. And then we can put them all
together with a couple of example problems. I told you
in the last video what it means to raise to a
negative power. a to the negative 1 power, or
maybe I should say a to the negative b power is equal to
1 over a to the b power. So just to do that with a couple
of concrete examples, 3 to the negative 3 power is equal
to 1 over 3 to the third power, which is equal to 1 over
3 times 3, times 3, which is equal to 1 over 27. If I were to ask you what 1/3
to the negative 2 power is-- well, this is going to be equal
to 1 over 1/3 to the second power. You get rid of the negative
and you inverse it. So this is going to be
equal to 1 over-- what's 1/3 times 1/3? 1/9. Which is equal to-- this is 1
divided by 1/9 is the same thing is 1 times 9, so
this is equal to 9. And this makes complete sense,
because 1/3, remember, 1/3 is the same thing as 3 to the
negative 1 power, right? 3 to the negative 1 is equal
to 1 over 3 to the 1 power, which is the same
thing is 1/3. So if we replace 1/3 with 3 to
the negative 1, this is 3 to the negative 1 to the
negative 2 power. These two things are equivalent
statements. And if we use one of the
properties we learned in the first video, we can
take the product of these two exponents. So this is equal to 3 to the
negative 1, times negative 2, which is just positive 2,
which is equal to 9. So it's really neat how all of
these exponent properties really fit together in a nice,
neat puzzle, that they don't contradict each other. And it doesn't matter which
property you use, you'll get the right answer in the end,
as long as you don't do something crazy. Now, the last thing I want to
define is the notion of a fractional exponent. So if I have something to a
fractional power-- so let's say I have a to the
1 over b power. I'm going to define this. This is going to be equal
to the bth root of a. So let me be very clear here. Let me make it with
some numbers here. If I said 4 to the 1/2 power
right there, this means this is equivalent to the
square root of 4. Which is equal to, if we're
taking the principal root, this is equal to 2. So if I were to take, let's be
clear, 8 to the 1/3 power, this is taking the
cube root of 8. And this is, on some level,
one of the most sometimes confusing things in exponents. Here I'm saying, what number
times itself 3 times is equal to 8 ? So if I said that x is equal to
8 to the 1/3 power, this is the exact same thing as
saying x to the third power is equal to 8. And how do I know that these
are equivalent statements? Well, I could take both
sides of this equation to the third power. If I take the left-hand side
of the third power and the right-hand side of the third
power, what do I get? On the left-hand side,
I get x to the third. On the right-hand side, I get 8
to the 1/3 times 3, which is just 3 over 3, which
is just 1. So if x is equal to 8 to
the 1/3, what is x? Well, 2 times 2, times
2 is equal to 8. And there's no really easy way,
especially once you go to the fourth root, or the fifth
root, and you have decimals of calculating these. You probably need a calculator
most of the time to do these. But things like 8 to the 1/3, or
16 to the 1/4, or 27 to the 1/3 , they're not too
hard to calculate. So this right here, let
me be clear, is 2. Now, let's make it a little
bit more confusing. What is 27 to the negative
1/3 power? Well, don't get too worried. We're just going to take
it step by step. When you take the negative
power, this is completely equivalent to 1 over 27
to the 1/3 power. These two are equivalent. You get rid of the negative
and take 1 over the whole thing. And then what is 27
to the 1/3 power? Well, what number times itself
3 times is equal to 27? Well, that's equal to 3. So this is going to be
equal to 1 over 3. Not too bad. Now I'm going to take it even to
another level, make it even more confusing, even
more daunting. Now, let me do something
interesting. What is 8 to the 2/3 power? Now that seems a little
bit scary. And all you have to remember
is this is the same thing, using our exponent rules
really, as 8 squared to the 1/3 power. How do I know that? Well, if I multiply these two
exponents, this is 2/3. So 8 to the 2/3 is the same
thing is 8 squared, and then the third root of that. But you could view
it the other way. This should also be equal to
8 to the 1/3 power squared. Because either way, when I
multiply these exponents, I get 8 to the 2/3. Let's verify for ourselves
that we really do get the same value. So 8 squared is 64. And we're going to take
that to the 1/3 power. Down here, we have
8 to the 1/3. We already figured
out what that is. That's 2, because 2 to
the third power is 8. So this is 2 squared. Now, what is 64 to the 1/3? What times itself 3 times
is equal to 64? Well, 4 times 4, times 4 is
equal to 64, or 4 to the third is equal to 64, which means
that 4 is equal to 64 to the 1/3. So this is equal to 4. And, lucky for us, 2 squared
is also equal to 4. So it doesn't matter which
way you do it. You could take the square and
then the third root, or you could take the third root
and then square it. You're going to get the
exact same answer. Now, everything I've
been doing has been with actual numbers. Let me do a couple of problems
that just bring everything we've done together
using variables. So let's say we wanted to do a
few expressions and we want to make sure there are no negative exponents in the answer. So let's add x to the negative
3 over x to the negative 7. There's a bunch of ways
we could view this. We could view this as equal to
x to the negative 3, times 1 over x to the negative 7. And what is 1 over x
to the negative 7? This is the same thing as x to
the seventh power, right? If you have 1 over something,
you can get rid of the 1 over, and put a negative in front
of the exponent. But if you're putting a
negative in front of a negative 7, you're going
to get x to a seventh. So this thing can simplify to
x to the negative 3, times x to the seventh power. And then we can add the
exponents, and that is x to the fourth power. Now, another way, a completely
legitimate way we could have done this, is we could have just
subtracted the exponents. We could have said, well, gee,
this is the same base. This is going to be x to the
negative 3, minus negative seventh power. Well, negative 3 minus negative
7, that's a negative 3 plus 7 which is equal to
x to the fourth power. And then one final way-- I mean,
actually, there's more than one final way we could
have done this. We could have said x to the
negative 3 over x to the negative 7-- sorry, not negative
x-- over x to the negative 7. Well, x to the negative 3 is the
same thing as 1 over x to the third-- that's that term
right there-- times 1 over x to the negative 7, so this would
have been equal to 1 over x to the third times
x to the negative 7. You could add the exponents,
so that's equal to 1 over 3 minus 7 is x to the
negative 4. And then this-- if we just get
rid of the inverse, we take the inverse of it, we can put
a negative in front of this negative, making it a positive--
this is going to be equal to x to the 4. So no matter how we did it, as
long as we're consistent with the rules, we got
x to the fourth. Let's do one more slightly
hairy one. And then I think we'll
be done for now. Let's say we have 3x squared
times y to the 3/2 power. And we're going to divide it by
x times y to the 1/2 power. Well, once again, this is the
same thing as 3 times the x terms right here, so 3 times x
squared over x, times y to the 3/2 over y to the 1/2. Well, this is going to be equal
to 3 times-- what's x squared over x? Or x squared over x to
the first power? That's going to be equal
to x to the 2 minus 1. And then this is going to be
times y to the 3/2 minus 1/2. So what does the whole
thing become? It becomes 3 times x. 2 minus 1 is just 1-- I can just
write x there-- times 3/2 minus 1/2 is 2/2. So that's y to the 2/2. 2/2, or 2 2ths-- that's just
the same thing is y. so this is equal to 3xy. Anyway, I encourage you
to do many, many more examples of that. But, you'll see that just using
the rules that we've been exposed to in the last
few videos, you can pretty much simplify any exponent
expression.