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My name's Miss Parnham.

In this lesson, we're going to learn how to simplify simple surds.

Let's recap some knowledge we already have about factor pairs of a number.

In this case, 80.

We can start with the most obvious pair of one and 80.

And then systematically work through until we found them all.

So obviously 80 is even, so two times 40.

It isn't a multiple of three, but it is a multiple of four.

It is a multiple of five, but not a multiple of six, not a multiple of seven.

It is a multiple of eight, not a multiple of nine.

And it is a multiple of 10, but we've already got 10, so we found them all.

These are all the factor pairs of 80.

The ones we are going to be interested in today are the ones that contains square numbers.

So in the case of 80, there's three of them.

we have one times 80, four times 20 and five times 16.

Now 16 is the largest square number factor of 80, and that's important.

And that's what we're going to be looking at Initially today.

Here are some questions for you to try.

Pause the video to complete the task and restart the video when you're finished.

Here are the answers.

Notice that in question two, I asked you to find the factor pair, containing the greatest possible square number.

This will be key to simplify answers.

Hopefully you've done the lesson on rational and irrational numbers.

And you know already that surd is a number written in square root form.

Let's start with a really simple multiplication and then convert the values to square root form.

Now these are not true surds but they're going to help us show something special about surds.

So we've converted to two to root four, three is root nine and six is root 36.

Now, do me a notice that for multiplied by nine is 36? Let's consider a further example to demonstrate that this is not just some coincidence.

So what's about five multiplied by four being 20.

If we convert those into square roots, then we have root 25 multiplied by root 16 is root 400.

And we notice again that 25 multiplied by 16 is 400.

So what we stumbled across here is a general rule.

Root of a multiplied by root of b is the root of ab.

And that is going to be so important to this lesson.

If you have a pen and paper, it might be worth writing that down and putting a box around it because that's so important for what we're about to look at.

Here's a quick multi-choice question for you to try.

Pause the video to complete the task and restart the video when you're finished.

Here are the answers.

Root 24 could be expressed as root eight, root three, or root two, root 12.

Simplifying the surds makes use of the rule that we've already looked at.

In this case, we know what ab is, and we need to find the best combination of a and b to simplify the surd as much as possible.

So let's think about root 40.

There are four ways that we can express root 40 as a product of two surds.

Only two of them contain a square number underneath the square root.

Root one is not going to make anything simpler because it is just one.

But we can exchange root four for two.

So root 40 can be written as two root 10.

We can apply that to other areas.

So here's a fraction with the denominator of root 40.

Let's exchange that for two root 10.

And here in this fraction, we can cancel common factors.

And the common factor is two.

So that simplifies to three over root 10.

The same is true of a ratio.

So root 40 is obviously root four, root 10.

And we can make that into two, root 10.

Looking at both sides of the ratio, we can, again, simplify by dividing by two, as it's a common factor.

So here we have root 10 to two root three.

And that's as simple as it goes.

Here's some questions for you to try.

Pause the video to complete the task and restart the video when you're finished.

Here's the answers.

In question four, you've probably spotted that the greatest square number factor of 96 is 16.

Therefore, if we wanted to simplify root 96, we would use root six, root 16, or root 16 root six, which is four root six.

Here are some further question for you to try.

Pause the video to complete the task and restart the video when you're finished.

Here are the answers.

Notice in part b of question six, it's initially simplify to root nine, root five, over root 16, root five, or three root five over four root five.

But the common factor of the denominator and numerator is root five and it will cancel leaving us with three quarters, which is a rational solution, and that's fine.

Here's some more questions for you to try.

Pause the video to complete the task and restart the video when you're finished.

Here are the answers.

Some of these simplified sets have common factors of the same surds, so cancelling them gives rational answers for both numbers in the ratio.

That's all for this lesson.

Thanks for watching.