### Video Transcript

In this video, we’re gonna be
multiplying radical expressions and simplifying the results. Radical expressions, remember, are
exact forms of numbers that involve radicals or root signs, like root five or seven
root three and stuff like that.

Our first question then is, express
five root five times root 45 in its simplest form.

Now, radicals in their simplest
form have the radicands, these bits here under the root signs, as small as they can
be after factoring out any square factors. So firstly, that five right up
against the root five means five times root five. And next, we’ve got to simplify
root 45. And now, we’re going to try and
find the biggest square factor we can find of 45. And the basic technique for doing
this is try dividing by two, then by three, then by four, then by five, and so
on. And see if any of the results that
we get are square numbers. And the first one we encounter is
also gonna be the biggest square factor. So 45 divided by two doesn’t
work. 45 divided by three is 15. But that’s not a square number. 45 divided by four doesn’t go into
45 exactly. 45 divided by five is nine. And that is a square number. So that is our largest square
factor. So I’m gonna rewrite 45 as nine
times five.

And now, root nine times five,
we’re gonna split out into root nine times root five. And the reason for doing that is
that root nine is three. Nine is a perfect square so root
nine is just three. Now with multiplication, it doesn’t
matter what order you multiply things together in. You’re gonna get the same
result. So I’m gonna swap around the root
five and three in the middle and multiply things together in this order. So we’ve got five times three is
15. And the square root of five times
the square root of five is just five. And 15 times five is 75, so the
answer is 75.

The next question then is express
four root seven times two root 49 in its simplest form.

Now, a couple of things when we
look at this, the four right up against the root seven means four times root
seven. And the two right up against the
root 49 means two times root 49. But also 49 is a perfect square, so
root 49 is exactly seven. So we can rewrite this as four
times root seven times two times seven. And again, with multiplication it
doesn’t matter what order you multiply things together in. You’re gonna get the same
answer. So I’m gonna to reorder those to
four times two times seven times root seven. And four times two is eight and
eight times seven is 56. So that’s 56 root seven. And if we look at seven, it’s — in
terms of its factors, only one and seven are factors. So obviously one is a square
number. But if we factor that one, it’s not
gonna enable us to make that radicand any smaller. So 56 root seven is our answer.

And the next question is, use the
distributive property of multiplication to expand five times three minus two times
the square root of five.

Now, the distributive property of
multiplication tells us that this means five times three take away five times two
root five. And five times three is 15. And five times two root five, well
that means five times two times root five. And five times two is 10, so that’s
10 root five. So we’ve got 15 minus 10 root
five. And that won’t simplify down any
further, so that is our answer.

Now, the next question is, simplify
three root three times two plus five root three.

So we’re gonna use the distributive
property of multiplication to do three root three times two. And then we’re going to add three
root three times five root three. Now, let’s rewrite that with all
the proper multiplication signs in there. And let’s swap things around a
little bit because it doesn’t matter what order you multiply things together in. So that gives us three times two
times root three plus three times five times root three times root three. Well, three times two is six. So this bit becomes six times root
three or just six root three. And three times five is 15. And the square root of three times
the square root of three is just three. So this gives us six root three
plus 15 times three. Well, 15 times three is 45. So it’s important to take note of
the order of operations. But this all simplifies down to six
root three plus 45.

Now, our next question is littered
with negative signs. So we’ve gotta be really, really
careful.

We’ve got to simplify negative
three root two times negative four take away two root two.

First of all, I’m gonna put
parentheses around the negative numbers. Just to make sure that we know that
they’re negative, and we don’t forget the negative sign. And then, we’re gonna use the
distributive property of multiplication to do negative three root two times negative
four. And then we’re gonna take away
negative three root two times two root two. Now, this bit here just means
negative three times root two times negative four. And because they’re multiplied
together, I can put them in any order I like. And likewise, with these terms
here, I’m going to put all the radical terms together and all the rest of the terms
together.

Right, let’s break down these
things and multiply things together. So I’ve got negative three times
negative four, which is positive 12. And that is multiplied by root
two. Then we’ve got negative three times
two, which is negative six. And the square root of two times
the square root of two, which is just two. So this overall right-hand
expression here becomes negative six times two, which is negative 12. So we’ve ended up with 12 root two
take away negative 12. Well, if we take away negative 12,
that’s the same as adding 12. And root two won’t simplify down
anymore, so this is our answer, 12 root two plus 12.

Next stop then, we’ve gotta
simplify two sets of parentheses multiplied together, root three plus five and root
three minus four. And to do that, we’re gonna
multiply each term in the first bracket by each term in the second bracket like
this. So that gives us root three times
root three. And then we got root three times
negative four. So we’re gonna be taking away root
three times negative four, which we’re gonna write the four before the root
three. So that becomes negative four root
three. Then we’ve got positive five times
root three, so that’s positive five root three.

And finally, five times negative
four, which is negative 20. And looking at those, we’ve got
root three times root three. Well that’s just three. Then we’ve got negative four of
these root threes. And then we’re adding five root
threes to that. So that’s gonna give us positive
one root three or just positive root three, as we could write it. And finally, we just got a negative
20 on the end. So combining like terms, we’ve got
three take away 20 is negative 17. And then, we’ve just got one root
three on its own. So the answer is negative 17 plus
root three. And like all of these questions, it
doesn’t matter actually which order you write those terms in. So you could write root three minus
17. In fact, there’s a case for doing
this because quite often we don’t like to lead with negative signs like that. So root three minus 17 is another
perfectly good answer.

Our next question then is seven
plus root five times seven minus root five.

Now, there are two different ways
we can approach this. So we’ll do both of them, and then
you can compare between the two. And the first method is to just
multiply each term in the second bracket by each term in the first bracket. And that gives us seven times seven
is 49. Seven times negative root five is
negative seven root five. Then root five times seven, well
I’m gonna do that the other way around. Seven times root five, so that’s
positive seven root five. And lastly, root five times
negative root five. Well, the square root of five times
the square root of five is just five. And positive times negative makes
negative, so that’s gonna be negative five. So we’ve got negative seven root
five plus seven root five. Well, they’re gonna cancel each
other out. So that’s gonna leave nothing in
the middle. So we’re just left with 49 minus
five, which is 44.

So multiplying out the parentheses
like that, not massively difficult. And it’s given us our simple answer
of 44. But let’s have a look again at the
question. If you recognise this form, that is
the difference of two squares. If you think back to your factoring
of quadratics, 𝑎 squared minus 𝑏 squared, so one square number take away another
square number, can be factored like this. 𝑎 plus 𝑏 times 𝑎 minus 𝑏. And if we let 𝑎 equal seven and 𝑏
equal root five, that would give us seven plus root five times seven minus root
five, which is what we’re trying to do. So this is just 𝑎 squared minus 𝑏
squared. In other words, seven squared minus
root five squared. Well, seven squared is 49, and root
five times root five is just five. So we’ve got straight to 49 minus
five, which is 44.

So as long you remember the rule of
two squares, and you keep that kind of stuff in your brain. Then this kinda question can
actually give you less working out to do this way then you had the other way.

Now, our next question, simplify
six root seven minus four root two times six root seven plus four root two.

And if we look inside these
parentheses, we’ve got six root seven in both places. And we’ve got four root two in both
places. So we’re subtracting in one, adding
in the other. This is just like the last
question. This is a difference of two squares
question. And this time, 𝑎 is six root
seven, and 𝑏 is four root two. So we can rewrite this as six root
seven squared minus four root two squared. And six times root seven all
squared means six times root seven times six times root seven. And four root two all squared means
four times root two times four times root two. And with multiplication, remember,
it doesn’t matter what order we multiply these things together. So I’m just gonna slightly
reorganise that.

So just swapping the order that I
multiply some of those terms together means that I’ve got the radicals together and
the rest of the numbers. So I’ve got six times six times
root seven times root seven and four times four times root two times root two. Well, six times six is 36, and root
seven times root seven is just seven. So I’ve got 36 times seven. And then, four times four is
sixteen. And root two times root two is
two. So this simplifies to 36 times
seven minus 16 times two. Now, I’m sure you’re all familiar
with your 36 times table. So 36 times seven is 252
obviously. And 16 times two is 32. Then, 252 minus 32, it gives us our
answer, 220.

Now, our next question, simplify
root five minus root six all squared.

Now, be very careful when you do
these sorts of questions. The most common mistake is just
people look at that especially in the heat of an exam and they think, “Oh, that’s
root five all squared minus root six all squared.” And so they end up doing five minus
six. Now that is completely wrong. You should always write it out in
full; something squared is that thing times itself. So root five minus root six all
squared is root five minus root six times root five minus root six. And we’ve got to multiply each term
in the second bracket by each term in the first bracket. So that’s root five times root
five, and then we’ve got root five times negative root six. That’s minus root five root
six. And then we’ve got minus root six
times root five.

Well, it doesn’t matter what order
you multiply them together in. So I’m gonna put that as root five
root six again. And it’s negative times positive,
so it’s a negative again. And then, we’ve got negative root
six times negative root six or negative times negative is positive. So looking at those expressions,
we’ve got root five times root five, which is just five. And we’ve got root six times root
six, which is just six. In fact, it’s positive six. Now, root five times root six is
also the same as root five times six. So we’ve seen this before when
we’ve been factoring out and simplifying these surds or these radicals. But it works the other way as
well. So root five times root six is root
five times six, and that is root 30. So we’ve got negative root 30 take
away another root 30.

Now let’s look at the terms;
collect like terms. Five plus six is 11 and negative
root 30 take away another root 30 is negative two root 30. Now looking at the contents of this
radical, 30. Let’s think about the factors of
30. Well, 30’s got lots of factors. But looking at them, only one is a
square number. And that’s not gonna help us in
this situation to reduce the size of the number. So we can’t factor it down any
further to simplify that radical or that surd. So this is our answer, 11 minus two
root 30.

Let’s move onto our second last
question then.

Root seven time three root seven
minus five minus two times four minus five root seven.

Well, we’re gonna use the
distributive property of multiplication to tackle this question. So looking at the first parentheses
first, we’ve got root seven times three root seven, which is the same as root seven
time three time root seven. Then, we’ve got root seven times
negative five. So we’re taking away root seven
times five which is the same as five times root seven. Then, we’re gonna do negative two
times four which is negative eight. So we’re taking away eight. And negative two times negative
five root seven. Well, negative two times negative
five is positive 10. So we’ve got positive 10 root
seven.

Right, let’s look through here. Now, in this first term here, we’ve
got root seven times root seven times three. Well, root seven times root seven
is seven, and seven times three is 21. And we can’t simplify the second
term, so minus five root seven. And we’ve got minus eight. And likewise, we can’t simplify the
last term. Now, 21 take away eight is 13. And negative five root seven plus
10 root seven is positive five root seven. And since seven doesn’t have any
square factors bigger than one, we can’t simplify this down any further. So 13 plus five root seven is our
answer.

Lastly then, just for a bit of fun,
let’s try this pretty tricky-looking question.

Simplify the square root of 15 plus
the square root of 19 all squared times the square root of 15 minus the square root
of 19 all squared.

Now, if you’ve been paying
attention throughout the video, you’re probably spotting the fact, “Oh, that looks
like difference of two squares!” But it’s not quite in that format,
is it? So the difference of two squares
says that 𝑎 squared minus 𝑏 squared is 𝑎 plus 𝑏 times 𝑎 minus 𝑏. And we’re nearly there. But we’ve got this problem of these
two squareds above the parentheses here. So if we do a bit of rearranging
then, we will be able to use that. So let’s have a look at it. Well, root 15 plus root 19 all
squared is just root 15 plus root 19 times root 15 plus root 19. So that’s the first parentheses
dealt with, and similarly for the second.

And now, we’ve got four sets of
parentheses, all multiplied together. And when we multiply things
together, it doesn’t really matter what order we do them in. So I’m gonna rearrange these. So I’ve just swapped around the
order of the middle two sets of parentheses there. And that’s left me with root 15
plus root 19 times root 15 minus root 19 all times root 15 plus root 19 times root
15 minus root 19. Now each of those is the difference
of two squares format that we were looking for. So if 𝑎 is root 15 and 𝑏 is root
19, we have this pattern here. And we also have it here. And that means that 𝑎 squared is
root 15 all squared, which is 15. And 𝑏 squared is root 19 all
squared, which is 19. And 𝑎 squared minus 𝑏 squared is
15 minus 19.

So let’s go back to our question
then. We’ve got 𝑎 plus 𝑏 times 𝑎 minus
𝑏 times 𝑎 plus 𝑏 times 𝑎 minus 𝑏. And 𝑎 plus 𝑏 times 𝑎 minus 𝑏 is
the same as 𝑎 squared minus 𝑏 squared. And as we just said, 𝑎 squared
minus 𝑏 squared is 15 minus 19. And 15 minus 19 is negative
four. So that becomes negative four times
negative four which is positive 16.

So remembering our difference of
two squares and doing a little bit of reorganising to get things in the right format
for that has saved us an awful lot of multiplying out. And got us to a very simple answer
of 16 instead of all of this stuff up here, which we started off with at the
beginning of the question.

Okay, good luck with multiplying
radical expressions then.