At this point
we should know how to simplify fractions and we should also
know how to simplify radicals. They're gonna get mixed up here, a little bit. Let
me show you.
Because, consider if you would,
two scientists doing a study. We'll say for the cure for cancer or something.
Okay now, one of them finds the correct dosage to be this value.
Okay, seems reasonable. It's a radical in a fraction.
Okay, whatever it comes out to. Now, another one
he says the correct dosage
is this value: square root of 3 over three. Okay.
So they both think they have a different answer and we want to compare
their answers and both answers appear to be in simplest terms because the
fraction can't be reduced
in either side, right? And the radicals are in lowest form.
Okay, well, actually we have another problem here.
Okay, consider if we wanted to compare their two answers.
Without getting a decimal,
Uh, what would we do? Consider
one thing. If we took the first guy's
answer and multiplied by the square root of three over the square root of 3. I do't want to tell you where
I got that but hold on just a second.
What would happen if I did and...
First of all, why can't I do it?
Because the square root of 3 over square root of 3 is one so I can do that anytime I want.
Okay why I did it will show you but I can certainly do it,
And if I do it what do I get?
Well, on the top I get square root of 3
and in the bottom I get square root of 9.
Well, that was the first scientis'ts
result and if I compare the two
the top is the same and actually
the bottoms are the same too.
So their answers, even though originally they look completely different,
their answers were the same and we didn't know it.
That's gonna present a problem. Even though both
answers were really the same our problem was
the two scientists wrote their answers differently.
Okay, the first guy put his radical in the denominator
and the other one put his radical in the numerator.
But we still want to be able to compare our answers
one with each other. We need to pick
one-way that we're all gonna do it. We call that a convention. So, the convention is:
From now on all radicals will be in numerators
only so that we can compare things.
Okay?
No radicals are allowed in denominators so you're gonna have to simplify any answer
or do whatever it takes
so that you do not have a radical in a denominator.
This process is called rationalizing
a denominator, getting the radicals out of our denominator.
Now, how do we do it? Show you how it's done.
Not so bad to remove a radical from a denominator.
Basically what we're gonna do is we're gonna multiply the numerator and the
denominator
by whatever root makes that denominator
a perfect root. Then he won't be a root anymore. For instance, in this case if I
multiply top and bottom
by radical two... Remember why I can do that?
As long as you multiply the top and bottom by the same number you're really
only multiplying by one and you're allowed to do that anytime
and this result will be three times radical two
is just three radical two and in the denominator
radical two times radical two is
radical four but he becomes a perfect square.
And you basically have removed the radical from the denominator.
Now it looks good.
Okay, now, we're allowed to have radicals in the numerator.
Okay? You just can't have them in the denominator so that is a correctly
simplified answer. Now there's a couple ways to look at the,
if you would, more difficult problems. For instance, this one.
Remember how I told you
multiply the numerator and the denominator certainly by the same thing
by whatever root
will make the denominator a perfect root. You don't always have to use the
denominator.
For instance, if I use the denominator here
Remember top and bottom!
of course the numerator would be a radical
and the bottom would be a big nightmare
but at least would be eighteen times 18 and the square root of that number would
be eighteen.
Okay, you think well, that wasn't too bad except that
I still have some more simplification to do because in the numerator
I've got a break it into good and evil square root of nine and two
and I get three and then I still have to simplify the fraction.
Good. Make my day! And I'll get a final answer
of radical two over six. Don't, don't even think of
canceling the two and the six because a radical doesn't cancel with a number.
And that would be one way to do it; that would be a long way
Let me give you an
easier way and then we said it from the beginning. Let's multiply by whatever it
takes to make that 18
into the next highest perfect square. What's the next highest perfect square
that 18 goes into is 36.
So if I multiply (cause I have to do it to the top and bottom -
whenever you do the top you have to do the bottom) acceptor
The top will give me
radical two; my bottom
will give you the perfect square, the smallest
perfect square and then
take the root. I think it's a much
easier way to do it; a little bit more thinking but
a lot, a lot less more computation. Okay.
How about this bad boy? Notice
it's a cube root. Remember what we told you.
Multiply both the numerator and denominator by whatever root it takes to make the
denominator a perfect root.
It's got to be a perfect cube root.
Okay, if you multiply top and bottom by the cube root of 3 over the cube root of
three...
while it is legal because it's one.,I don't think that's gonna help you.
Because the top will be cube root and the bottom,
will be the cube root of nine
which is not a perfect cube. We have to make it... In this case it's a
whole different ballgame, a perfect cube root. I really think the process we told
you at the beginning
is the better way to go. What's the next biggest cube
that you could multiply? What it's gonna take to multiply to make that bottom
into a perfect cube?
I smell 27. What does it take to get to 27?
I'm gonna have to multiply the top and bottom by whatever it takes to get that
to be 27 because that is a perfect cube.
It's going to be nine. A top is allowed
to have a root in it and we create a perfect
root in the bottom and then we take it...
And that's the better process.
I promise it'll work for all roots, okay?
Remember our goal was to get the route out of the denominator.
Now get to work.
Go do your homework.