How to Rationalize a Denominator

How to Rationalize a Denominator

In this video, we're going to
learn how to rationalize the

denominator.

What we mean by that is, let's
say we have a fraction that

has a non-rational denominator,
the simplest one

I can think of is 1 over
the square root of 2.

So to rationalize this
denominator, we're going to

just re-represent this number in
some way that does not have

an irrational number
in the denominator.

Now the first question
you might ask is,

Sal, why do we care?

Why must we rationalize
denominators?

And you don't have to
rationalize them.

But I think the reason why
this is in many algebra

classes and why many teachers
want you to, is it gets the

numbers into a common format.

And I also think, I've been
told that back in the day

before we had calculators that
some forms of computation,

people found it easier to have
a rational number in the

denominator.

I don't know if that's
true or not.

And then, the other reason
is just for aesthetics.

Some people say, I don't
like saying what 1

square root of 2 is.

I don't even know.

You know, I want to how
big the pie is.

I want a denominator to
be a rational number.

So with that said, let's learn
how to rationalize it.

So the simple way, if you just
have a simple irrational

number in the denominator just
like that, you can just

multiply the numerator and the
denominator by that irrational

number over that irrational
number.

Now this is clearly just 1.

Anything over anything or
anything over that same thing

is going to be 1.

So we're not fundamentally
changing the number.

We're just changing how
we represent it.

So what's this going
to be equal to?

The numerator is going to be 1
times the square root of 2,

which is the square root of 2.

The denominator is going to be
the square root of 2 times the

square root of 2.

Well the square root of
2 times the square

root of 2 is 2.

That is 2.

By definition, this squared
must be equal to 2.

And we are squaring it.

We're multiplying
it by itself.

So that is equal to 2.

We have rationalized
the denominator.

We haven't gotten rid of the
radical sign, but we've

brought it to the numerator.

And now in the denominator we
have a rational number.

And you could say, hey, now I
have square root of 2 halves.

It's easier to say even,
so maybe that's another

justification for rationalizing
this

denominator.

Let's do a couple
more examples.

Let's say I had 7 over the
square root of 15.

So the first thing I'd want to
do is just simplify this

radical right here.

Let's see.

Square root of 15.

15 is 3 times 5.

Neither of those are
perfect squares.

So actually, this is about as
simple as I'm going to get.

So just like we did here, let's
multiply this times the

square root of 15 over the
square root of 15.

And so this is going to be equal
to 7 times the square

root of 15.

Just multiply the numerators.

Over square root of 15 times
the square root of 15.

That's 15.

So once again, we have
rationalized the denominator.

This is now a rational number.

We essentially got the radical
up on the top or we got the

irrational number up
on the numerator.

We haven't changed the number,
we just changed how we are

representing it.

Now, let's take it up
one more level.

What happens if we have
something like 12 over 2 minus

the square root of 5?

So in this situation, we
actually have a binomial in

the denominator.

And this binomial contains
an irrational number.

I can't do the trick here.

If I multiplied this by square
root of 5 over square root of

5, I'm still going to have an
irrational denominator.

Let me just show you.

Just to show you
it won't work.

If I multiplied this square root
of 5 over square root of

5, the numerator is going
to be 12 times the

square root of 5.

The denominator, we have
to distribute this.

It's going to be 2 times the
square root of 5 minus the

square root of 5 times the
square root of 5, which is 5.

So you see, in this situation,
it didn't help us.

Because the square root of 5,
although this part became

rational,it became a 5, this
part became irrational.

2 times the square root of 5.

So this is not what you want
to do where you have a

binomial that contains an
irrational number in the

denominator.

What you do here is use our
skills when it comes to

difference of squares.

So let's just take a
little side here.

We learned a long time ago--
well, not that long ago.

If you had 2 minus the square
root of 5 and you multiply

that by 2 plus the square root
of 5, what will this get you?

Now you might remember.

And if you don't recognize this
immediately, this is the

exact same pattern as a minus
b times a plus b.

Which we've seen several
videos ago is a

squared minus b squared.

Little bit of review.

This is a times a, which
is a squared. a

times b, which is ab.

Minus b times a, which
is minus ab.

And then, negative b
times a positive

b, negative b squared.

These cancel out and you're
just left with a

squared minus b squared.

So 2 minus the square root of 5
times 2 plus the square root

of 5 is going to be equal to
2 squared, which is 4.

Let me write it that way.

It's going to be equal to 2
squared minus the square root

of 5 squared, which is just 5.

So this would just be equal to
4 minus 5 or negative 1.

If you take advantage of the
difference of squares of

binomials, or the factoring
difference of squares, however

you want to view it, then
you can rationalize this

denominator.

So let's do that.

Let me rewrite the problem.

12 over 2 minus the
square root of 5.

In this situation, I just
multiply the numerator and the

denominator by 2 plus the square
root of 5 over 2 plus

the square root of 5.

Once again, I'm just multiplying
the number by 1.

So I'm not changing the
fundamental number.

I'm just changing how
we represent it.

So the numerator is
going to become 12

times 2, which is 24.

Plus 12 times the square
root of 5.

Once again, this is
like a factored

difference of squares.

This is going to be equal to 2
squared, which is going to be

exactly equal to that.

Which is 4 minus 1, or we
could just-- sorry.

4 minus 5.

It's 2 squared minus square
root of 5 squared.

So it's 4 minus 5.

Or we could just write that
as minus 1, or negative 1.

Or we could put a 1
there and put a

negative sign out in front.

And then, no point in even
putting a 1 in the

denominator.

We could just say that this is
equal to negative 24 minus 12

square roots of 5.

So this case, it kind of did
simplify it as well.

It wasn't just for the sake
of rationalizing it.

It actually made it look
a little bit better.

And you know, I don't if I
mentioned in the beginning,

this is good because
it's not obvious.

If you and I are both trying to
build a rocket and you get

this as your answer and I get
this as my answer, this isn't

obvious, at least to me just by
looking at it, that they're

the same number.

But if we agree to always
rationalize our denominators,

we're like, oh great.

We got the same number.

Now we're ready to send
our rocket to Mars.

Let's do one more of this, one
more of these right here.

Let's do one with
variables in it.

So let's say we have 5y over
2 times the square

root of y minus 5.

So we're going to do this
exact same process.

We have a binomial with an
irrational denominator.

It might be a rational.

We don't know what y is.

But y can take on any value, so
at points it's going to be

irrational.

So we really just don't want a
radical in the denominator.

So what is this going
to be equal to?

Well, let's just multiply the
numerator and the denominator

by 2 square roots of y
plus 5 over 2 square

roots of y plus 5.

This is just 1.

We are not changing the
number, we're just

multiplying it by 1.

So let's start with
the denominator.

What is the denominator
going to be equal to?

The denominator is going to
be equal to this squared.

Once again, just a difference
of squares.

It's going to be 2 times
the square root of y

squared minus 5 squared.

If you factor this, you would
get 2 square roots of y plus 5

times 2 square roots
of y minus 5.

This is a difference
of squares.

And then our numerator is 5y
times 2 square roots of y.

So it would be 10.

And this is y to the first
power, this is

y to the half power.

We could write y square
roots of y.

10y square roots of y.

Or we could write this as y to
the 3/2 power or 1 and 1/2

power, however you
want to view it.

And then finally, 5y times
5 is plus 25y.

And we can simplify
this further.

So what is our denominator
going to be equal to?

We're going to have 2
squared, which is 4.

Square root of y squared is y.

4y.

And then minus 25.

And our numerator over here is--
We could even write this.

We could keep it exactly the
way we've written it here.

We could factor out a y.

There's all sorts of things
we could do it.

But just to keep things simple,
we could just leave

that as 10.

Let me just write
it different.

I could write that as this is
y to the first, this is y to

the 1/2 power.

I could write that as even
a y to the 3/2 if I want.

I could write that as y to
the 1 and 1/2 if I want.

Or I could write that as 10y
times the square root of y.

All of those are equivalent.

Plus 25y.

Anyway, hopefully you found
this rationalizing the

denominator interesting.