### Video Transcript

Let’s have a look at simplifying
some radicals. Now depending where you live, you
might use terms like irrational numbers or surds. But the specific aspect of this
that we’re going to be looking at today is rationalising denominators. So if we’ve got expressions like
this one one over the square root of two the denominator, the “downstairs”, of that
fraction has got a radical or it’s got a surd number just sitting there: the square
root of two. Now that looks like quite a simple
number to most of us. But within mathematical circles,
we don’t like that situation where you’ve got those radicals on the denominator, so
we have to try and eliminate those. And that’s really what we gonna be
looking at in this video.

So in our first example,
rationalise the denominator of one over the square root of five. Okay so let’s just write out one
over root five equals one over root five. Well that seems pretty obvious;
it’s equal to itself. But let’s try this then, so one
over root five times one. If we times anything by one then
you’re gonna get the same value. So that’s clearly true as well. Now we’re gonna look at different
flavours of one different versions of one.

Okay, so root five over root five,
something divided by itself will give you the answer one, so clearly root five over
root five is one. So we’ve done the same thing here;
that is still one over root five times one. Now the reason I picked root five
over root five is because look the denominator here contains root five; I know that
root five times root five is just five. And five is not a radical number;
it’s not a surd; it’s not in root format; it’s a- it’s a rational number. So on the numerator, one times
root five is just root five, and root five times root five is five.

So there’s our answer. Now, you know this is supposed to
be simplifying these radicals; and in fact, this-this answer here and probably looks
a little bit more complicated than this answer here. This has is got smaller digits in
it, smaller and numbers on the numerator and the denominator. But nonetheless, to simplify a
surd or to simplify a radical in this format, we want to eliminate these radicals
from the denominator. So that’s the answer that we’re
looking for.

So let’s look at another example
then: four over root twelve. So the basic process is gonna be
just what we did before, but there might be a little bit more cancelling down to be
done at the end as well. So as before, we said four over
root twelve is equal to four over root twelve times one, and the version of one that
we’re going to be using is — yep, you guessed it. It’s gonna be root twelve over
root twelve so that we can eliminate the radical from the denominator.

So there we have that, root twelve
over root twelve is just one, so all we’ve done is we’ve taken the fraction that we
were given and we multiplied it by one. So we’re not changing the
magnitude, the size, of this fraction. So multiplying those two fractions
together, we’ve got one big fraction four times root twelve on the top and root
twelve times root twelve on the bottom. So remember, root twelve times
root twelve is just twelve.

And the numerator is still four
root twelve four, four times root twelve. So we’ve got four times root
twelve on the numerator and twelve on the denominator. Well look four and twelve, they’ve
got a common factor of four. So four divided by four is just
one; twelve divided by four is three. So I’ve cancelled that down a
little bit and simplified it.

So we can write that out as root
twelve over three. Now actually, that’s not our
answer because given that we’re trying to — we have rationalised the denominator, so
technically we have answered that question. But typically this would be
simplify- fully simplified this radical; and in that case, we’ve got a way of
simplifying the numerator a little bit further as well. Now the temptation is to cancel
the three with the twelve. But remember because it’s root
twelve on the top, you can’t do that because it’s not root three on the bottom, so
we can’t do any cancelling there. But twelve can be written as four
times three, and four is a square number so it’s a square factor of twelve.

And root of four times three is
the same as the square root of four times the square root of three, and of course
the square root of four is two. So that gives us two root three
over three. So we’ve just illustrated there is
that the- when we’re simplifying these surds, these radicals, we always looking to
remove radicals from the denominator. But we’re also looking to make the
contents of these square roots, the number under the root if you like, as small as
it possibly can be. And by factorising out this square
factor of twelve, four and then evaluating that the square root of four is two,
we’ve managed to make this number, this value, simpler as we would say in
mathematical terms.

Well, sometimes the fractions are
a little trickier.

So in this question we’ve been
asked to fully simplify fifteen plus root three, which is on the numerator, over
root three. Now world’s number one mistake in
terms of these things is just cancelling off the root threes and getting the
question wrong. But you can’t do that; remember,
you can only cancel factors. And we’ve got fifteen plus root
three on the top, not fifteen times root three, so we can’t cancel down those
factors like that. Okay, let’s look at what we can
do. We’ve got a root three on the
denominator. So if I multiply that fraction by
one, but the version of one I’m gonna use is root three over root three.

So I bracketed it together that
fifteen plus root three just to make sure that we know that they’re together. And I’m gonna multiply the whole
of that numerator by root three and I’m gonna multiply the whole of the denominator
by root three, remembering that root three divided by root three is one, so we’re
just multiplying by one. Okay, so the reason we’ve done
that is because when we multiply these two things together, root three times root
three, it just gives us three. That’s gonna eliminate our radical
from the denominator.

So now all I’ve got to do is
multiply out the terms of the numerator, so root three times root three is three,
and root three times fifteen is fifteen root three. Now again we’re gonna look to see:
can we factorize? Can we cancel? What can we do? Now if we look at that numerator
there, I’ve got fifteen root three plus three. Well three is a factor of three
and three is also a factor of fifteen. So I could simplify that numerator
a bit by factorising it.

And three lots of five root three
is fifteen root three and three lots of one make three over here. So I’ve just factorised the
numerator there. Now it’s probably worth mentioning
now that I’ve got three times the whole of five root three plus one and I’ve got
three on the bottom. So now we can cancel things out,
so three is a factor of three. If I divide three by three, I get
one. If I divide three by three I get
one. So I’ve got one lot of five root
three plus one all over one. Well clearly.

I don’t need to have the multiply
one and divide by one in there. I can just simplify that to five
root three plus one or one plus five root three. Doesn’t matter which way you write
those two round.

Okay, let’s look at this example
then.

Fully simplify one over root two
plus one. So this time we’ve got a slightly
more complicated term on the denominator rather than the numerator, and this does
make life a little bit more difficult. Our general approach is just the
same we’re going to multiply that by some version of one. But in this case the version of
one that we gonna multiply by is root two minus one. So what you do for these ones, you
have a look at the denominator. So we’ve got root two plus one and
you just change the sign, and that gives us the term that we’re going to use to
multiply those out. You’ll see why that is in a moment
when we actually do the multiplication. For those of you are thinking
ahead a little bit, think difference of two squares and what happens when you use
the difference of two squares if you know about that. Okay, so when we multiply out the
numerator I just got one lot of that whole bracket so that’s gonna to be fairly
straightforward.

So that pa- next one, one lot of
root two is positive root two and one lot of negative one is just negative one. So looking at our denominator,
we’ve got the square root of two times the square root of two, which is just two. We’ve then got negative root two
add root two. So if we start off at negative
root two and we add it to itself, that’s just gonna give us zero. So those two things cancel out, so
we’ve got two take away one.

And two take away one is just one,
so we’ve got negative root- sorry! we’ve got root two take away one all over one. Well we don’t need to write over
one. So the whole thing just simplifies
down to the square root of two take away one.

Right then, one more example, so
fully simplify nine over three minus the square root of three. Now it’s always a good idea to put
denominators in brackets; if you got a couple of terms, then bracket them together,
same again with the numerators so it’s clear which terms have to stick together. And we’ve got to find a version of
one to multiply this by which is gonna completely get rid of all those radicals from
the denominator in this fraction. Now what we said before is we look
at it. We’ve got the term here; we’ve got
three; we’ve got negative root three here. So we’re gonna — if we change that
sign, it would give us three plus root three. That’s the term which we’re gonna
put on the numerator and the denominator to make our version of one.

So in order to simplify this, what
we’re gonna do is a nice complicated process where we multiply the top and the
bottom of that fraction by three plus root three. So I’m not actually gonna multiply
out the numerator just yet because there’s a chance, you know, with these questions
that maybe if we factorise something later on, something might cancel out. So sort of save us a little bit of
work, I’m not really gonna do anything for now with the numerator. But let’s look at the denominator,
I’ve gotta do three times three, which gives us nine. I’m gonna do three times positive
root three so they’re both positive, so it’s gonna be positive answer so positive
three root three.

Then moving on to the next terms,
we’ve got negative root three times three, so that’s negative three root three, and
then negative three root three times positive root three. Well negative times positive’s
gonna give us a negative, and root three times root three is three. That’s the definition of square
roots.

So let’s see if we can tidy up
that denominator a little bit. I’ve got nine and I’m taking away
three, so that’s just six. And I’ve got positive three root
three and then I’m taking away three root three fr- I’m taking my three root three
from itself. So they’re gonna cancel out and
give us zero, three root three take away three root three is zero. So there’s nothing else to put on
that denominator.

Now I’ve got nine times three plus
root three on the top and I’ve got six on the bottom. So because I’ve got things
multiplied together, factors, I can do some cancelling. three is a factor of nine
and of six, so six divided by three would be two, nine divided by three will be
three. So that gives us our answer three
lots of three plus root three all over two. Not really a lot of cancelling I
can do there now; that is in fact my answer.

Now if I multiplied out that
numerator and got nine plus three root three all over two, that is just as good an
answer; that’s not a problem. And in fact, I could even split
that out into two separate fractions nine over two plus three root three over two. All equivalent answers and all of
them are correct.

So when we talk about
rationalising denominators, it just means getting rid of all these kind of square
rooty type things from the denominator, and the way that we do it, remember, is
multiply by some version of one, which is gonna help us to wipe out those Radicals
from the denominator.