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# Evaluating fractional exponents

Video transcript

We've already seen how to
think about something like 64 to the 1/3 power. We saw that this is
the exact same thing as taking the cube root of 64. And because we know
that 4 times 4 times 4, or 4 to the third
power, is equal to 64, if we're looking for
the cube root of 64, we're looking for a number that
that number times that number times that same number is
going to be equal to 64. Well, we know that number is 4,
so this thing right over here is going to be 4. Now we're going to
think of slightly more complex fractional exponents. The one we see here has
a 1 in the numerator. Now we're going to see
something different. So what I want to do is think
about what 64 to the 2/3 power is. And here I'm going to use
a property of exponents that we'll study more later on. But this property
of exponents is the idea that-- let's say
with a simple number-- if I raise something
to the third power and then I were to raise that
to, say, the fourth power, this is going to be the same
thing as raising it to the 2 to the 3 times 4 power, or 2 to
the 12th power, which you could also write as raising it to
the fourth power and then the third power. All this is saying is, if I
raise something to a power and then raise that
whole thing to a power, it's the same thing as
multiplying the two exponents. This is the same thing
as 2 to the 12th. So we could use
that property here to say, well, 2/3 is the
same thing as 1/3 times 2. So we could go in
the other direction. We could say, hey
look, well this is going to be the same
thing as 64 to the 1/3 power and then that thing squared. Notice, I'm raising
something to a power and then raising
that to a power. If I were to multiply
these two things, I would get 64 to the 2/3 power. Now, why did I do this? Well, we already know what
64 to the 1/3 power is. We just calculated it. That's equal to 4. So we could say that--
and I'll write it in that same yellow
color-- this is equal to 4 squared,
which is equal to 16. So 64 to the 2/3
power is equal to 16. The way I think
of it, let me find the cube root of 64, which is 4. And then let me square it. And that is going
to get me to 16. Now I'll give you in
even hairier problem. And I encourage you to
try this one on your own before I work through it. So we're going to
work with 8/27. And we're going to raise
this thing to the-- and I'll try to color code
it-- negative 2 over 3 power, to the negative 2/3 power. I encourage you to pause
and try this on your own. Well the first
thing I do whenever I see a negative
exponent is to say, well, how can I get rid of
that negative exponent? And I just remind myself, well,
the negative exponent really just says, take the
reciprocal of this to the positive exponent. I'm using a different color. I'm going to use that
light mauve color. So this is going to
be equal to 27/8. I just took the reciprocal
of this right over here. It's equal to 27/8 to
the positive 2/3 power. So notice, all I did, I
got rid of the exponent and took the reciprocal of
the base right over here. 8/27 is the base. Negative 2/3 is the exponent. Now, how can we handle this? Well, we've already
seen that if I have a numerator to some
power over a denominator to some power-- and this is
another very powerful exponent property-- this is going to be
the exact same thing as raising 27 to the 2/3 power-- to
the 2 over 3 power-- over 8 to the 2/3 power. This is another very
powerful exponent property. Notice, if I have something
divided by something and I'm raising the
whole thing to a power, I can essentially
raise the numerator to that power over
the denominator raised to that power. Now, let's think
about what this is. Well just like we
saw before, this is going to be the same thing. This is going to be the same
thing as 27 to the 1/3 power and then that squared
because 1/3 times 2 is 2/3. So I'm going to raise
27 to the 1/3 power and then square
whatever that is. All this color
coding is making this have to switch a lot of colors. This is going to be
over 8 to the 1/3 power. And then that's going to be
raised to the second power. Same thing we were doing
in the denominator-- we raise 8 to the 1/3
and then square that. So what's this going to be? Well, 27 to the 1/3 power
is the cube root of 27. It's some number-- that
number times that same number times that same number is
going to be equal to 27. Well, it might jump out at you
already that 3 to the third is equal to 27 or that 27
to the 1/3 is equal to 3. So the numerator, we're going
to end up with 3 squared. And then in the
denominator, we are going to end up with-- well,
what's 8 to the 1/3 power? Well, 2 times 2 times 2 is 8. So this is 8 to
the 1/3 third is 2. Let me do that
same orange color. 8 to the 1/3 is 2, and then
we're going to square that. So this is going to
simplify to 3 squared over 2 squared, which is just
going to be equal to 9/4. So if you just break
it down step by step, it actually is not too daunting.