Lesson video

Hi, welcome to today's lesson on simplifying surds.

By the end of today's lesson, you'll be able to fully simplify a surd.

Now there are two key words we'll be using today or perhaps key phrases is better and these are product of primes and simplest form.

Now you will have heard both of these expressions before so what I'd like you to do is to pause the video and write down what you think these mean, do this now.

Welcome back.

Let's see if your definitions match mine.

Expressing a number as a product of primes means we write the number uniquely as a product of its factors that are prime numbers.

For example, if I wanted to write eight as a product of its primes, I would write two times two, times two because all of those factors there are prime numbers.

If for example, I wanted to write 10 as a product of its primes, I would write 10 is equal to 2 times 5.

A surd is in its simplest form when the radicand remember, that's the value underneath the radical sign or the root sign.

So where the radicand is an integer with no perfect square factors greater than one and that's what we'll be looking at today.

In our previous lesson we were identifying the highest square factor, so we're gonna go further now and actually simplify some surds.

There are two parts to our lesson today and we're going to start with the first part on simplifying surds.

A surd is in its simplest form when the radicand is an integer with no perfect square factors greater than one.

Remember, the square root of 32 we looked at that last lesson is not in its simplest form because 16 is a factor of 32 and 16 is a perfect square.

Conversely, the root of 21 is in its simplest form because there are no perfect square factors of 21 that are greater than one.

To begin to simplify a surd, you must identify the highest square factor greater than one of the radicand.

Now, it's okay if you identify a square factor and it's not the highest.

Rather like with fractions we were simplifying if you could find the highest common factor rather than simply a common factor then you simplified in the minimum number of steps.

The same is true here with surds, if you can identify the highest square factor, then you are going to simplify as efficiently as possible.

If there are no square factors greater than one of the radicand, then your surd will be in its simplest form.

Here's our example again from earlier.

Root 32 is equivalent to saying it's the root of 16 times 2.

So that's not in the simplest form.

Conversely, the root of 21 is.

Now surds can have a coefficient.

The coefficient is not what determines whether or not the surd is in its simplest form.

It is still the radicand that determines this.

Which of these surds do you think are in their simplest form? Pause the video now and make your selection.

Welcome back.

Which ones did you choose? If you chose A and D, then you are correct.

B is no good because four is a factor of eight and four is a square number so B can't be right and we already knew that C root 32 is not in its simplest form because 16 is a square factor and 16 is a factor of 32 because 16 is a square factor of 32.

It's now time for your first task.

Now what's lovely about this is that you can have so many different correct answers here so there's a lot of room for creativity.

What you need to do is if possible find a surd that would go in each region that you can see here in our diagram.

So the circle at the top left is the group of surds that can be simplified.

In other words, to go within that circle, you must have a surd where the radicand has a perfect square factor.

The circle on the top right is the set of surds that when you multiply them by themselves the result is a multiple of three.

You might think quite carefully about that one.

And the bottom circle is when you evaluate that surd you have a value less than five.

Now this is all right because we know that when we square root 25 we get 5.

So if our radicand is smaller than 25 then we should have a surd that evaluates to something that is less than five.

But again, don't hesitate to use your calculator here to help you work out where surd can go.

If you're looking at one of the regions that is created by the intersection or overlapping of two circles then it needs to satisfy the conditions of both circles for it to go there and if it's in that central area, it has to be a surd that can be simplified, gives a value less than five and when you square it, it's a multiple of three so it's got to meet all three conditions.

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

Welcome back.

Let's see how you got on.

Now, please bear in mind that mine are just examples.

You may have different numbers to me, but it still works.

Here's what I've gone for.

Everything that's in my can be simplified circle all have values where the radicand has a perfect square factor.

All of the values that are in the circle on the top right when you square them are multiples of three.

In the bottom circle, all of these surds evaluate to something less than five.

Did you see the root of 101 sitting outside all the circles? That's because if I square root to 101 I get a value that is less than five.

If I square it, then I get 101 which is not a multiple of three.

And actually, that surd cannot be simplified because 101 does not have any perfect square factors that are greater than one.

Remember, you could have chosen different values here but still have got it right.

Let's look at simplifying surds.

Hmm, I've got a square here.

What length would go where the marked arrow is? Well, we know that a squares area is found by multiplying the length by the width and because it's a square, these are the same thing.

So what multiplied by itself gives me two? Perhaps a better way to think of it is I need to work backwards.

Something squared is two so square rooting two will let me work out what that answer is, oh but the square root of two is an irrational number.

Root two is a surd.

Luckily, I know it's okay that I can be exact and I can write that as my length, so if I've got a square with a length of root two, then its area would be two.

Well, that seems straightforward.

What's about if I had a bigger square? Let's consider taking four of these and placing them as you can see in the diagram.

I know because I've made four copies of this little square that each of the little squares lengths when I put them together is still root two which is what you can see here.

Now, what would happen if instead of considering these as smaller squares, I considered the total square? In other words, if I've got four squares each of area two then the square made by putting four of them together has an area of eight.

Well, let's consider what the length of that square is.

I can see across the top I've got root two and root two.

Well, they were the two lengths that only went halfway.

Altogether, how many lots of root two do I actually have? Well, I've got two lots of root two.

Except I can think about my length in a different way.

What squared gives me a result of eight? Or in other words the square root of eight is the length of one side.

My diagram has just shown me that the square root of eight is the same as two multiplied by the square root of two.

These are equivalent forms. I know they have to be because I'm dealing with a square here and a square has four sides of equal length, so root eight is equivalent to two root two.

The root eight was not simplified because it has a highest square factor of four.

Two root two, conversely is simplified because there are no perfect square factors of two that are greater than one.

Let's consider that again.

The square root of eight can be written as the square root of four multiplied by the square root of two.

In other words, two root two.

Let's try writing the square at 45 in its simplest form using the same reasoning.

Pause the video now and have a go at doing this.

Welcome back.

Did you use the diagram to help you? You could have done if you did, you would've seen that each little square has an area of five and for the large square we have nine lots of five, which is 45 so the area of the large square is 45 which means that the length of the large square must be the square root of 45.

Can you see using the length along the top of the square if we sum those smaller lengths we can see what root 45 is equivalent to.

That's right, it's equivalent to three root five.

We can write the square root of 45 as the square root of 9, the highest square factor multiplied by 5.

We can then say, well, the square root of nine is three leaving us with three root five.

Now we're going to give you some key facts to help with simplifying surds.

We know that when we square root a perfect square we get a lovely rational answer, so square to 36 is 6.

When I square root and value squared those inverse operations cancel out and I'm just left with the radicand from my original surd.

Square root of nine we know is three and therefore the square root of three squared is also three.

In general, when we square root a number that is squared we get back to the original number.

For now, that's a fact we're going to be using.

Later on in our lessons, we're actually going to derive this fact.

This is another one that we're going to use.

Root three multiplied by root four is the same as saying three to the power of a half multiplied by four to the power of a half, and we know this for our work on indices because the exponent of the same we can write the basis as being multiplied together.

And therefore we can write 12 to the power of half or the square root of 12.

In general, multiplying two values that are both square rooted is the same as square rooting the product of those two values, or in other words the square two A multiplied by the square of B is the same as the square of A times B.

We'll be exploring that in more depth when we come to general multiplication of surds later on in this unit.

But for now we're using this result to help us simplify surds.

Let's look at how we'd write the square of 72 in its simplest form.

Well what is the highest square factor of 72? I know it's 36 so I'm going to rewrite 72 as 36 multiplied by 2.

I can then break this up so it's the root of 36 multiplied by the square root to 2 or it's the root of 6 squared multiplied by root 2 so 6 multiplied by root 2 or 6 root 2.

This is now fully simplified.

Remember, identifying the highest square factor is an efficient way to simplify a surd.

Of course, we could write 72 as a product of its prime factors.

Can you see what I've done here? I've rewritten two cubed to be two times two squared because I have two squared times three squared I could write that as two times three or squared.

I could then say well this is root to multiplied by the root of six squared or root two times six, six root two.

So I can use either method for identifying the highest square factor to simplify my surds.

Let's try this with root 75.

I'd like you please to fill in the gaps to complete this working to fully simplify the surd.

Pause the video and do this now.

Welcome back.

We know that 75 is equal to three lots of 25 or three times five squared.

We can then break this up so we have root three multiplied by the square root of five squared and therefore five root three.

It's now time for our second task.

I'd like you in part A to fully simplify the two surds that you see there.

Remember, try to identify the highest square factor first or you'll need to keep simplifying.

In part B, I'd like you to unsimplify each surd so there's no longer a coefficient and I've given you an example.

Three root two would be the same as saying the square root of nine multiplied by the square root of two because three squared gives me nine and therefore the square root of nine must be three and root 9 times root 2 is the square root of 18.

Pause the video while you complete this task.

Welcome back.

It's now time for the final part.

Lucas, Izzy and Alex have been fully simplifying the surd root 432.

One of them has fully simplified it.

Unfortunately, the others have made mistakes.

What mistakes do you think they have made? Pause the video while you complete this.

Welcome back.

Let's go through our solutions.

So for a, I know that 12 can be written as 4 times 3.

The square of four is two, so this gives me a result of two root three.

272, I've gone for writing it as a product of its primes here, can be written as 2 to the power of 4 multiplied by 17.

Well, two to the power of four can be written as two squared or squared without listing with 2 squared times root 17 or 4 root 17.

You might have worked out that it was 16 times 17 and gone straight to 4 root 17 that way, and that's fine too.

In b, I asked you to unsimplify.

Well, two squared is four, so the square to four must be two so I'd have root four multiplied by root 6 which is the square root to 24.

For 2 root 15, well, 2 squared is 4 so that's root 4 times root 15 or root 60.

And then for four root five, the four becomes the square root of 16, so the square root of 16 times a square root 5 gives me a surd but is the square root of 80.

Now who fully simplified that surd? That was Izzy.

Izzy's done a great job there.

432 is the same as saying 144 multiplied by 3.

So Izzy has then square to the 144 to get 12 so she ends up with a result of 12 root 3.

Lucas didn't square root the 144, if he had have done he would've got to 12 root 3, just like Izzy.

Alex has done good work.

What he's done is he's spotted that there was a factor of 432 and he's used that, but he got to 4 root 27, he can actually simplify further.

27 can be written as 9 times 3 so actually he'd then end up with three root three.

He's already multiplying by four so he'd get to the 12 root 3 like Izzy but it's a lot harder for him.

It's now time for part two and this is where we simplify surds with a coefficient and this is where Alex had gotten to.

Let's see how we do this.

So similar to when working with algebraic terms when a surd is written three root two it means three times root two.

That makes sense, it's what we saw in the first part of today's lesson.

Now this has implications when we simplify surds that have coefficients.

We're going to write 2 root 18 in its simplest form.

That form will be something multiplied by root two whether something is an integer.

Let's see what happens.

2 multiplied by the root of 18 is equal to 2 lots of the square root of 2 times 9.

Well, we can break that up, remember so that's two times root two times root nine except root nine is equivalent to three.

Can you see now I have two integer values I can multiply together? The two multiplied by the three gives me six so this becomes six root two.

In other words, when we simplified our surd, we then remembered to multiply by the already existing coefficient.

It's now your turn.

I'd like you please to write 4 multiplied by root 27 in its simplest form.

Now that will be whether radicand is three and you'll have a coefficient that's an integer.

Pause the video and do this now.

Welcome back.

Let's see how you got on.

Well, we can write 27 as 3 squared times 3 or you might have written 9 times 3.

We can then break this up so it's four times the square to three squared times by root three.

Well, this just leaves us with 4 times 3, which is 12 times by root 3, so we have 12 root 3.

It's now time for your final task.

I'd like you to match each surd with its simplest form.

Now, you'll notice that in each of the simpler forms either the coefficient is unknown or the radicand is unknown.

When you've matched the surds up, tell me what that unknown is, so what's its value? Pause the video while you complete this task now.

Welcome back.

Let's see how you got on.

The first one I started with was this one because I realised that 36 times 6 is equal to 216.

Or 36 square root is 6, so I'd end up with 36 root 6.

And there was no coefficient there that already said 36 and no other radicand that said 6 so it must be this one and P must be 36.

I then looked at the 7 root 40 and said, well, root 40, well 40 is 4 times 10 so the square at 40 will simplify to 2 root 10 but I've already got seven as my coefficient so I'm going to need to do 7 times 2 and that will give me 14 root 10 and I could see there were no other coefficients that were 14 or radicands that were 10 so it must match to the top one here and X must be 14.

I then did 2 root 60 and realised that's simplified to 4 root 15 because 60 remember can be written as 4 times 15, so root 60 becomes 2 root 15 multiplied by the existing 2, 4 root 15.

This then left me with just 2 root 45 and the root of 180.

I've matched them this way, so the square of 180 can be written as 36 times 5.

Square root of 36 is 6, so 6 root 5.

Now, I could have matched that one of course to the one that just above because I had R root five written there, but even if I did R would still be six and likewise with 2 root 45 when I simplify that, root 45 becomes the root of 9 times 5 or 3 root 5 multiplied by 2 gives you 6 root 5.

In other words, hints exactly the same.

The important bit is whichever way round you've matched it R must be six and A must be five.

It's now time to summarise what we've learned today.

A surd is in its simplest form when the radicand is an integer with no perfect square factors greater than one.

And to simplify a surd efficiently we need to identify the highest square factor.

We also used these two facts with surds and we're going to explore them in more depth in our future lessons.

Well done, you've worked really well today and I can see you putting a lot of effort.

I look forward to seeing you for our next lesson.