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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.