Lesson video

Hi, welcome to today's lesson on multiplication of surds.

By the end of this lesson, you'll be able to appreciate the structure that underpins the multiplication of surds.

Our lesson today has three parts and we're going to start with part one, which focuses on squaring a surd.

Now you can see six statements on the screen.

What I'm asking you is are these statements true or false? Show some working to justify your answers.

You can do these all without the aid of a calculator, but by all means, feel free to check your reasoning afterwards using your calculator.

Pause the video now while you have a go at this.

Welcome back.

Let's see what you put.

Well, for the first one, the square root of 200 is equivalent to 10 root two.

We know that's true because 200 can be written as a hundred multiplied by two and the square root of a hundred is 10.

So we know that simplifies.

Now what about the second statement? The square root of nine, subtract the square root of four is equal to the square root of five.

That's right, it's false.

We know that in order to sum surds, we need the radicands to be the same and they're not here at all.

In fact, we can simplify that by saying that's actually equivalent to three takeaway two which would be one, not the square root of five.

Likewise for the surd one we could actually say that that reads as three add two.

Is five equal to the square root of 13? No, of course it's not.

It's false.

Now what about these three sets of statements? Is the square root of four multiplied by the square root of four equal to 16? That's right.

It's definitely not.

In fact, underneath we've got the square root of four multiplied by the square root of four yet again.

Only this time we're saying it's equal to four, and that is indeed true.

The square root of four is just two.

So two multiplied by two is indeed four, not 16.

And what about the last one? The square root of nine multiplied by the square root of nine.

Is that equal to three? No, it's not.

The square root of nine is three.

So actually what we're saying there is three multiplied by three is three.

No, it's nine.

We're going to look at what happens when we square a surd and that's what the second part of that exercise was focusing on.

Let's consider multiplying the square root of three by itself.

Well, we know from our work on indices that we can write this in a different way.

The square root can be written as an exponent of a half.

So root three multiplied by root three can be written as three to the power of a half multiplied by three to the power of a half.

Now we know from our work on the laws of indices that there's an equivalent expression we could write here.

Because the exponents are the same, we can multiply their bases together.

So what we've actually got is three multiplied by three all to the power of a half.

Well, let's simplify what's inside the bracket.

In other words, it's equivalent to nine to the power of a half.

But remember when the exponent is a half, that's equivalent to square rooting.

So actually we're saying that's the square root of nine.

The square root of nine of course is three.

Look what happened when we multiplied a surd by itself, ie, when we squared it.

It makes sense that we got back to what our radicand was.

Try this one using the same reasoning.

Can you tell me what the value of the square root of 11 multiplied by the square root of 11 is? You may already have an idea, but make sure you can justify it with some working.

Pause the video while you do this now.

Welcome back.

Let's just check you have the same working as me.

Remember, we can write this as 11 to the power of a half multiplied by 11 to the power of a half.

We can then combine the bases because the exponent is the same for both and we are multiplying.

This reaches 121 to the power of a half, which is the same as saying the square root of 121, which is 11.

Can you see what happened there? What about if we wrote it like this? Well, the square root of three all squared is the same as saying the square root of three multiplied by the square root of three and by the reasoning we just saw that's equivalent to three.

It's now your turn.

Find the value of the square root of 11 all squared.

Pause the video and write down your answer.

Welcome back.

Did you write 11? Well done if you did.

Again, it's the same reasoning as before.

This brings us to our first task.

I'd like you to find the value for each of the questions you see below.

In some of them we've written out the multiplication and the others, we've gone for the shorter form of just squaring.

Write down what the value is in each case.

Pause the video now.

Welcome back.

Let's see how you got on.

Well for A, B, C, D, and E, it was straightforward.

What you had there was simply the radicand in each time.

We know from our reasoning in our checks just before this task that when we multiply a surd by itself we're simply left with whatever the radicand was.

This applies to F, G, and H as well, in that the notation just means multiply the surd by itself.

Did you spot that I and J though, were slightly different? In I we're asking you to multiply root three over two by root three over two.

In other words, we're multiplying two fractions together.

We know when we multiply two fractions that the numerators are multiplied, so root three times root three will give us three and so are the denominators.

In other words, two times two gives us four.

Did you spot that you also had to square the two? Well done if you did.

In J, we have two root three all squared.

It can help to write this out.

So we would've written two root three multiplied by two root three.

Two times two is four and root three times root three is three.

So we have four times three, which gives us 12.

It's now time for the second part of our lesson and in this we explore the structure of multiplying surds.

We saw that squaring a surd resulted in a value equal to the radicand.

What do you think will happen though, when we multiply together two surds with different radicands? For example, root eight multiplied by root two? Let's see what does happen.

We are going to show that root eight multiplied by root two is the same as the square root of 16.

Now that might seem a little unusual because we know that the square root of eight is an irrational number and so is the square root of two but yet their product is going to produce a rational answer.

The square root of 16 after all is four.

Let's see how we can justify this.

We know that the square root of eight can be simplified to be written as two root two.

Now all I've done is just pointed out that when we see a coefficient by a surd, we know it means to multiply.

So our statement actually reads as two multiplied by root two multiplied by root two.

Hang on a second.

We just did multiplying a surd by itself.

We know that's therefore equivalent to two multiplied by two, which is is the same as four, but four can be written as follows.

Four is of course the square root of 16.

Can you see how we wrote root eight multiplied by root two as the square root of 16? It's now your turn.

I'd like you please to show that root 12 multiplied by root three is equivalent to the square root of 36.

You can use the reasoning on the left hand side of the screen to support you here.

Pause the video and have a go now.

Welcome back.

Let's see how your reasoning got on.

First of all, you should have taken root 12 and written it as simply as possible.

Root 12 can be written as two root three.

So we have two lots of root three multiplied by another lot of root three.

We know that's the same as two times root three times root three.

root three multiplied by root three is just three.

So we have two times three.

Two times three is six and six is the square root of 36 which is what we were required to show.

What do you think will happen here? So I'd like you to find the value of root 50 multiplied by root two.

You can use your previous reasoning to support you working out your answer here.

Pause the video and have a go.

Welcome back.

What did you put? That's right, the answer is 10.

You could have written that it's the square root of a hundred.

However, you weren't asked to give your answer in a particular form this time, and we should always try to give the simplest form of any answer.

What will happen in this case? Well, anything squared means that I must multiply it by itself.

So I'd have root six multiplied by root two, multiplied by root six multiplied by root two.

Remember, multiplication is commutative.

IE, it doesn't matter whether we do root two multiplied by root six or root six multiplied by root two.

The end result is the same.

Root six multiplied by root six is just six and root two multiplied by root two is just two.

So we've actually got six multiplied by two or 12.

Now what that means is is that root six multiplied by root two must be equal to root 12.

What we can conclude is that when multiplying surds, we multiply the radicands together.

Let's complete the calculation to show that root seven multiplied by root three is the same as the square root of 21.

The first line of working's been done for you.

You need to fill in the remaining three lines so that you can reach the required result.

Pause the video and do this now.

Welcome back.

Let's check your lines of working.

So the first thing you had to do was use the Commutative law to swap the middle two terms in our multiplication.

We then had root seven times root seven gives us seven, and root three times root three gives us three, meaning that seven times three is 21 and then by square rooting we know that root seven times root three is equal to root 21 as required.

Or in other words, we multiplied the radicands together to get our results.

What would be the value of root two multiplied by root 13? You don't need to do all the working in between now if you feel confident and can just put down the answer.

Pause the video and do this now.

Welcome back.

Did you put the square root of 26? Because two multiplied by 13 is 26.

What is the value of root three multiplied by root six? Welcome back.

Did I manage to catch you out? How many of you put A? You could have done, but that's not a simplified answer.

So if you've put A, I caught you because you're meant to fully simplify.

18 can be written as nine times two and therefore root 18 simplifies to three root two.

It's time for your second task.

In this, I'd like you to match the multiplications you see on the left with the simplified answers on the right.

Remember, I've simplified these answers so you may not initially get the answer you are expecting.

Don't forget to simplify and you should find it appearing in the list on the right.

Pause the video and do this now.

Welcome back.

Let's see how you got on.

Root six times root 8 is simply root 48 but the square root of 48 can be written as the root of 16 multiplied by three or four root three.

Four root five multiplied by root five is just four times five, so that's 20.

The square root at 15 times the square root of five would be the square root of 75.

Well, 75 can be written as 25 times three.

So we simplify to five root three.

And then root six multiplied by root five is simply root 30.

No simplifying there because there are no square factors of 30 that are greater than one.

It's now time for the final part of our lesson today and that's the structure of dividing surds.

So we've looked at multiplying, it's now time for division.

Division is of course the inverse operation of multiplication.

If I have root two multiplied by root three, we know that the result is now root six because we would multiply the two and the three together to produce the radicand we need for our product.

Now knowing the statements of multiplication, we can rewrite these as two division facts.

Because root two multiplied by root three is root six, I know that root six divided by root two must be equal to root three.

I can also write that root six divided by root three must be equal to root two.

Can you spot what's happening here? When dividing surds, we are dividing the radicands.

Using the statement root 12 multiplied by root five is equal to the root 60, fill in the two statements about division that you can see below.

Pause the video and do this now.

Welcome back.

Let's see what you filled in.

Both statements should start with root 60 because that was the product when we were multiplying.

So we should have root 60 divided by root 12 is equal to root five, and root 60 divided by root five is equal to the root of 12.

It's now time for your final task and in it we're going to do a little bit of practise with multiplication and division.

I'd like you to calculate and then fully simplify please.

Pause the video and do this now.

Welcome back.

It's now time for the second part.

In this part of the task, I'd like you please to find the scale factor of enlargement.

In other words, the rectangle on the right was created from the rectangle on the left using a scale factor.

You could of course argue that the rectangle on the left was created from the rectangle on the right using a scale factor.

And of course it was.

The important bit is that these two rectangles are similar in that there's a scale factor used to multiply both sides to produce the other rectangle.

I'd like you to find this scale factor please and then find for me the length of the missing side in the smaller rectangle.

Pause the video and do this now.

Welcome back.

Let's see how you got on by starting with part A.

Square root of 75 divided by the square root of 15.

Well, that's 75 divided by 15 is five so my answer must be root five.

When I had the root of 48 divided by the root of six, well, 48 divided by six gave me eight but root eight is not in the simplest form.

The square root of eight can be written more simply as two root two.

Now in the third one, I had the square root of an eighth multiplied by the root of 216.

Did you spot that this is equivalent to doing a division? 216 divided by eight, gave me 27.

27 can be written as nine multiplied by three and therefore simplifies to three root three.

In the last one, I'm doing the root of 108 divided by the root of a sixth.

Well, that means I'm doing 108 divided by a sixth, or 108 multiplied by six, and this then simplifies to 18 root two.

In B, we said first of all you should find the scale factor of enlargement.

So what I did was I said, well, the root five and the root 35, they're connected by a scale factor.

What did I multiply root five by to get to root 35? Well, it must've been root seven, and that makes sense because 35 divided by five is equal to seven.

So root seven is my scale factor.

To work out the length of A therefore, I can take the side in the larger rectangle, root 21, and then divide that by root seven to find the missing side in the smaller rectangle.

Square root of 21 divided by the square root of seven is equal to the square root of three and that must be the value for A.

It's now time to summarise what we've done in our lesson today.

The square root of A all squared is equal to A.

In other words, when we square a surd, we're simply left with the radicand.

When multiplying surds, we multiply the radicands together and when dividing surds, we divide the radicands.

Well done.

You've worked really well today and put in a lot of effort.

I look forward to seeing you in our next lesson.