 # Evaluate an expression with Rational Exponents without a Calculator

In this video, I’ll show you how
to evaluate an expression with a rational exponent, like
we have here, without a calculator. We have 27 to the four
thirds power. So first off, let’s take a look
at that exponent, four thirds, and break it up into
its individual parts. For this fraction, we have two
parts as we do in every fraction, the numerator
and the denominator. First up, the bottom,
the denominator. Here it’s a 3. For an exponent, this is going
to tell us what type of root we’re going to take. Meaning it’s a radical,
and this is the index for that radical. Since this is a 3, we’re going
to want to take a cube root at some point. If it had been a 2, it would
be a square root. If it were 7, you’d have
a seventh root. Whatever number you have in the
denominator becomes the index for the radical. Now, the top. The numerator here, the 4, acts
more like a traditional exponent, or how you’d
be more comfortable dealing with exponents. And it’s basically the power
that we’re going to raise this to. If we raise something to the
fourth power, we’re going to take it and multiply
it four times. So ultimately, both of these
things have to happen. Something is being raised to
the fourth power and we’re taking the cube root
of something. Now the question just becomes,
what order do we want to do it in? Well, if we look at the number
four thirds, we can recognize the fact that it’s equal to 4,
that power, times 1/3, what we used to indicate the root. And if we write it in this
order, this would mean that we would take the power first,
and then take the root. So for this example, we would
end up having the number 27 raised to the fourth power, take
that quantity and have the cube root of it. Now, remember, we’re trying to
do this without a calculator. And it’s entirely possible
to take 27 and multiply it four times. So 27 times 27 times 27 times–
yep, 27 and do that without a calculator. I can pretty much guarantee you
though, if I tried to do it now in this video, I would
end up screwing it up and doing it wrong somehow. So trying to multiply that large
of a number that many times is somewhat inviting
making mistakes, because everyone makes them at some
point or another. Furthermore, that number that
we end up with will be huge. And the next thing we’d have to
do is to take the cube root of that number, again without
a calculator. So though this is possible, it’s
taking a small number, or relatively small number, 27,
first making it huge by doing a lot of multiplication, and
then trying to take a very difficult cube root
after that. So instead of doing it this
way, it might be a little easier to think of this in the
reverse order and think of 4/3 as 1/3 times 4. Meaning in the order we want
to do it here, we’re first going to take that cube root. Then, raise that result
to the fourth power. This means that we’re going
to take 27, make it into a smaller number by taking the
third root, and only have to multiply that result
four times. Since we’re doing this without
a calculator, that will be a slightly easier way to do it. And then, just this slid right
out of my way here. Sorry about that. Let me move this up and still
keep things on the screen, and show you how this works out. So again, my original problem
is 27 to the four thirds. And we said the easiest way to
do this in terms of keeping the numbers as small as
possible, and therefore as easy to work with as possible,
is to first take the root. Meaning raise 27 to the
one third power. Take that amount to
the fourth power. By the rules of exponents, we
know if you have an exponent here raised to another exponent
here, you end up multiplying the exponents. And 1/3 times 4 is 4/3. Now, we’re going to rewrite
it in its radical form. Remember, the 27 to the 1/3
is the cube root of 27. It’s just another way to
write that same idea. This is going to be raised
to the fourth power. Well, the cube root of 27 is
3, which means instead of trying to multiply 27 four
times, I’m only going to be multiplying 3 four times. This gives me 3 times 3 times
3 times 3, which is 81– my final answer. So without a calculator, and
just by doing this in the easiest order possible, we can
calculate 27 to the four thirds power and get 81.