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CCSS Math: HSN.RN.A.2, HSN.RN.A

Fractional exponents can be
a little daunting at first, so it never hurts to do as
many examples as possible. So let's do a few. What if we had
25/9, and we wanted to raise it to the 1/2 power? So we're essentially
just saying, well, what is the principal
square root of 25/9? So what number times
itself is going to be 25/9? Well, we know 5 times 5
is 25, and 3 times 3 is 9. So why don't we
just go with 5/3? Because notice, if you
have 5/3 times 5/3, that is going to be 25/9. Or another way of saying
this, that 5/3 squared is equal to 25/9. So 25/9 to the 1/2 is
going to be equal to 5/3. Now let's escalate
things a little bit. Let's take a really hairy one. Let's raise 81/256 to
the negative 1/4 power. I encourage you to pause this
and try this on your own. So what's going on here? This negative-- the first
thing I always like to do is I want to get rid of this
negative in the exponent. So let me just take
the reciprocal of this and raise it to the positive. So I could just say that
this is equal to 256/81 to the 1/4 power. And so now I can say,
well, what number times itself times
itself times itself is going to be equal
to 256, and what number times itself times
itself times itself-- did I say that four times? Well, what number, if I take
four of them and multiply, do I get 81? And one way to
think about it, this is going to be the
same thing-- and we'll talk about this in
more depth later on when we talk about
exponent properties. But this is going to be
the exact same thing as 256 to the 1/4 over 81 to the 1/4. You, in fact, saw it over here. This over here was the same
thing as the square root of 25 over the square root of 9. Or 25 to the 1/2
over 9 to the 1/2. So we're just doing
that over here. So one, we still have to think
about what number this is. And this is a little bit of,
there's no easy way to do this. You kind of have
to just play around a little bit to come up with it. But 4 might jump out at you if
you recognize that 16 times 16 is 256. We know that 4 to
the fourth power, or you're about to know this,
is 4 times 4 times 4 times 4. And 4 times 4 is 16, times 4
is 64, times 4 is equal to 256. So 4 to the fourth
is 256, or we could say 4 is equal to
256 to the 1/4 power. Fair enough? Now what about 81? Well, 3 might jump out at you. We know that 3 to
the fourth power is equal to 3 times 3 times 3
times 3, which is equal to 81. So 3 is equal to 81 to the 1/4. So this top number, 256
to the 1/4, that's just 4. 81 to the 1/4, that is just 3. So this right over here is
going to be equal to 4/3.