Rational Exponents 1

>> This is part one of rational exponents.
So, remember the word rational basically means fractions.
That's how you want to think of it.
So what we're going to do is define fractional exponents.
So far we've defined exponents for integers,
for instance b cubed means b times b times b. B
to the zero equals one, b to the negative two is one
over b squared, which is one over b times b. All of these are
in my videos on exponents and I have a lot on those.
So now we're going to go on to,
what about if we have a fractional exponent?
Well before doing that, let's just look at these two laws
of exponents: b to the m times b to the n is b to the m plus n
and also b to the m raised to the nth power is b to the mn
because we'll be using those.
Let's say I wanted to know what five
to the one-half was equal to.
Okay, so I want to know what five
to the one-half is equal to.
Well notice that if I multiply five to the one-half times five
to the one-half, using my laws of exponents, I write the base
down and then I add the two exponents.
One-half plus one-half is just one so basically five
to the one-half times five to the one-half is five
to the first power, which is just five.
So what I want you to think about is,
what number times itself is going to equal five?
And that would be the square root
of five times the square root of five.
Alright, let's look at it a different way.
What if I had five to the one-half, I'm trying to figure
out what five to the one-half equals, right?
And so if I square that, if I took five to the one-half
and squared it, I would have five
to the - what do we do here?
We multiply exponents.
One-half times two is one, again I'd have five.
So the question is what number squared equals five?
And again from our rules with radicals,
that would be the square root of five, squared, equals five.
Hmmm, so what do you think five to the one-half equals?
Well, it's going to have to be the same thing as square root
of five for this to make sense.
So, we have that five
to the one-half equal's square root of five.
And in fact, if you have any number to the one-half,
it really is just going to be the square root of that number.
So if I had sixteen to the one-half that would mean what?
That would mean the square root of sixteen.
And what's the square root of sixteen?
It's four.
Again, remember that we take the principle square root,
the non-negative one.
Alright, let's look at ten to the one-third.
Let's do a similar thing.
If I have ten to the one-third, what happens if I multiply it
by itself three times?
You would have the base and add one-third plus one-third plus
one-third is three-thirds, or one.
So, it just equals ten.
Hmm. Similarly, I could have written that as ten
to the one-third cubed, right?
That's what that means.
Ten to the one-third cubed also, using my laws of exponents,
one-third times three, I would have ten
to the first power, which is ten.
Right? So interestingly enough, I'm thinking,
"Well what else number cubed equals ten?"
What's that going to be?
Hmm, that's got to be the cube root of ten.
Remember? Cubing, taking the cube root,
sort of un-does each other, so the cube root of ten, cubed,
is going to equal ten.
So you know what?
Ten to the one-third is the same thing as cube root of ten.
What would, let's see, how about sixty-four
to the one-third equal?
That would mean the cube root of sixty-four, and so you have
to think what number cubed is 64, and that's going to be four.
Now when I did it for the square roots, we actually didn't want
to right the two in here, but notice if it's
to the one-third power, I put the three in the radical.
For the square roots it's assumed
that there's a two in there.
So now let's go to the definition on b to the one
over n. If n is a positive integer, greater than one,
so those are the ones we're going to deal with right now.
And the nth root of b is a real number, so we're only talking
about real numbers, then b to the one
over n is simply the nth root of b. Now for the square root -
I'm sorry, I want to write two, so if I want to have b
to the one-half, we just right the square root
of b. You don't have to put the two in there, right?
If you want, you could put the two in there, that's up to you.
It's assumed.
So this b to the one-half could also be written
as the square root of b. You don't need to write
that little two in there.
Alright, so let's have you try these problems.
See if you could do these on your own,
if it's not a real number write "Not a real number".
Be careful of all your laws of exponents that you already know
and we're just doing an add-on.
What are you going to do with these fractional exponents?
So put the video on pause and try these on your own first.
Okay, let's do it.
Eighty-one to the one-half, what's that mean?
That means the square root of eighty-one.
And what's the square root of eighty-one?
Nine. Alright, next one.
Negative nine to the one-half.
Now I've written a parentheses around negative nine
so that's the base, so I want the square root
of negative nine.
And what's that?
Oh, it's not a real number.
Remember we're working with real numbers for the moment.
We will learn how to deal with the square root
of negative nine much later on.
Alright, how about negative twenty-seven to the one-third?
The minus sign is not in parentheses
so we are not taking the cube root of negative twenty-seven.
The minus sign is out in front,
the base is only the twenty-seven.
Okay, so the base is only twenty-seven so that's all
that goes in the cube root.
So we're taking negative the cube root of twenty-seven.
And that will be negative three.
Okay, hmm.
We have a fractional exponent and a negative number.
So the first thing I have to deal
with here is the negative sign with the exponent.
And remember what that means?
One over sixteen to the one-half.
So that's one over - now what's sixteen to the one-half?
That's the square root of sixteen.
So I have one over - what's the square root of sixteen?
Four, so this is one-fourth.
Tricky one.
And the last one here, five x to the one-fourth.
Just like number three, the five is not being raised
to the one-fourth power, that's like five to the first power.
So the five is sort of like the co-efficient and x
to the one-fourth means the fourth root of x. And that's
as far as you could go here because we don't know what x is.
So we can't simplify this one any further.
Alright, so there it is.
So you now know what a fractional exponent is,
here's the definition and I just want to make sure you realize -
notice that the denominator of the fractional exponent,
so when you have b to the one over n, that n what's
in the denominator here ends up being the index of the radical,
what goes inside that little radical part.
And of course with square roots, we don't write the two in there.
All right so the next video will go over fractional exponents
where there's not just a one in the denominator.
So we might have something like eight
to the two-thirds, etcetera.