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.