Lesson video

Hi, welcome to our lesson today on addition with surds.

By the end of today's lesson, you'll be able to appreciate structure that underpins the addition of surds.

Our lesson has three parts.

And we're gonna start by identifying what like surds are.

How might you group the surds you see on the screen? Have you got some ideas of what a group might look like? What might go where? What goes with what? Let's think about one way you could group them.

Surds that have the same radicand are like surds.

Remember the radicand is what appears under the root sign.

These are all examples of like surds.

We can see that the radicand is 15 in each case.

The coefficient of the surd, that can differ, but it's the radicand being the same that's important to identify our like surds.

These are not like surds.

We can see that we have 15, 2, 5 and 42.

Are these examples of like surds? Well, initially you might think no.

Because 12 and 3 are not the same.

Seems straightforward, right? Except this is where simplifying comes in.

Root 12 can be written as the square root of 4 multiplied by 3.

Remember we look at identifying the highest square factor.

Well, that's 4.

And now we can simplify that surd to become 2 root 3.

Ah, wait a second.

They are like surds.

But we couldn't see that until we'd fully simplified, and then the radicand was the same.

So when it was written as root 12 and root 3, they were not like surds.

It wasn't until I fully simplified that we reached a point where we did have like surds.

We can now identify these like surds from our grid.

Pause the video and have a quick go.

Welcome back.

How did you get on? Let's see if you spotted the like surds.

I'm gonna start with these ones.

So that's where the radicand was 2 in each case.

These are one group.

A group of like surds.

Let's remove them and see what other groups we can make.

Well, I can also see this group of like surds.

Again, did you spot that root 12? We know that simplifies to 2 root 3, which is why I've put it in this group.

'Cause when I fully simplify, it will have the same radicand.

Hmm, what about now? Is there another group we can see? That's right.

It's where ab is the radicand.

And what about now? It's right, there's just two left.

The root 6 I'm afraid is all by itself.

Like surds have the same radicand.

To determine if two surds are like surds, you may need to manipulate either one or both surds.

Fully simplifying will always work, but it may not be as efficient.

Let's consider fully simplifying both root 12 and root 8.

Well, we know 4 is the highest square factor in both cases, and that leads us to 2 root 3 and 2 root 2.

But these are not like surds.

Although the coefficient is the same in both, the radicand is not.

So these are not like surds.

But it took us fully simplifying both to be able to determine that.

What about this case? Now we might be thinking we should always fully simplify both surds in order to determine if they're like.

Except that's not always necessary.

Let's consider 2 root 128.

We can write 128 as 4 multiplied by 32.

This then simplifies to 4 root 32.

And here we can see that these are like surds.

So I could have fully simplified and gone further, but I didn't actually need to.

There's no point doing more work than we have to after all.

Time for a quick check.

Identify which of the surds that you can see on the screen are the like surds.

Pause the video and have a go now.

Welcome back.

Did you pick these ones? We know that the root of 8 can be written as 2 root 2, and therefore, these are like surds.

What about these ones? Pause the video and have another go.

Welcome back.

Did you pick these ones? You can see for the bottom two, they clearly have the same radicand of b.

The top right one though, we know that a squared can be simplified to become a, so that we'll end up with -2a root b.

In other words, the same radicand.

It's now time for your first task.

I'd like you to identify the odd one out from each row.

Then write another surd that pairs with the odd one out.

In other words, it's a like surd.

Pause the video while you do this activity.

Welcome back.

Let's see how you got on.

The ones that weren't like surds have been circled for each of the rows.

Now at the end I've given an example that you could use that pairs with it, but you don't have to do the same as me.

For example, if you look at the top row and the bottom row, you'll see that I kept the radicands exactly the same as the one that was the odd one out.

In other words, no simplifying required.

For the second row though, I wrote root 40 because that's equivalent to 2 root 10.

so it is a like surd.

It does pair with the one that was odd, but it didn't have to be exactly the same radicand as long as it's simplified to be the same.

It's now time for the second part of our lesson and this is where we'll be adding like surds together.

Remember surds that have the same radicand are like surds.

Because the surd represents a unique value, these like surds can be collected together, exactly the same way as we collect like terms in algebra.

For example, 3 lots of a plus 4 lots of b minus 6 lots of a plus 7 lots of b gives us a total of -3a plus 11b.

Remember to simplify an expression means that we will collect all the like terms together.

There's our algebraic expression before and there's it simplified.

Let's consider what this would look like if we use surds in place of a and b.

So here we have the same expression, but instead of 3a, I have 3 root 2, and instead of 4b, I have 4 root 3.

In other words, I've substituted, and in place of a, I've written root 2, and in place of b, I've written root 3.

And look, the simplified expression is exactly the same form where, instead of -3a, I have -3 root 2, and instead of 11b, I have 11 root 3.

In other words, it's exactly the same as when we were simplifying with algebra.

It's time for a quick check.

I'd like you to identify the correct simplification of 3 root 5 take away root 5 add 3 root 3 add 2 root 5 add 7 root 3.

Remember to gather the like terms and then choose.

Is the correct simplified expression, A, B, or C? Pause the video and make your selection now.

Welcome back.

Which one did you pick? You should have chosen C.

If we look at our expression and we consider just the like terms that have 5 as the radicand, we'll have 3 take away 1, that's 2, add 2 makes 4.

So we should have 4 root 5.

The only expression that contains 4 root 5 is C.

So already we know it's going to be that one.

But we should check still by making sure we reach 10 root 3 as well.

Well, by considering the terms that have root 3 as part of them, 3 root 3 plus 7 root 3 is 10 root 3.

So we know for definite it's C.

It's now time for second task.

And what you need to do in this one is to fill in the gaps you can see on the screen so that each side is equivalent.

In other words, if it's for the left-hand side, which term or coefficient is missing so that it equals the right-hand side? And then for c and for d, it's a little bit more evolved in that you've got a missing term on both the left and the right for c, and in d, there's quite a lot there that you need to simplify by adding those like terms together.

Pause the video and do this now.

Welcome back.

Let's see how you got on.

Well let's start with a.

5 root 10 take away something root 10 is going to make -7 root 10.

Well we can just focus on the coefficients here.

5 take away what makes -7? We must be taking away 12.

In b, we know that in order to gather the like terms, the radicands must be the same.

So we know that root 7 will be part of our answer.

What do I add to 6 to make 9? 3.

So 3 root 7.

In c, there was quite a lot going on here.

We'll focus, first of all, on the terms containing root 2.

3 root 2 plus 7 root 2 is 10 root 2, then take away 4 lots of root 2 will leave us with 6 root 2.

And that makes up the answer box that's on the right.

We now need to deal with the terms involving root 3.

We can see that we're going to get to -11 root 3.

On the left-hand side, we can see there's a -6 root 3, but we need to get to -11, so we must have to take away 5 lots of root 3.

Because -5 take away 6 makes -11.

In d, we simply had to simplify what was on the left-hand side.

Now in this case, we can see that we have numbers that don't involve surds.

So we'll start with those first.

4 take away 16 is -12.

Well -2 root b plus 2 root b gives us a result of zero.

Now you could write plus zero, but there's actually no point is there.

So we don't need to put that in our answer.

Now let's consider the terms where the radicand is bc.

We have 7 root bc take away root bc, that leaves us with 6 lots of root bc.

It's now time for the final part of our lesson and this is where we need to simplify before we can add the like terms together.

Surds that have the same radicand are like surds.

Surds with different radicands cannot be summed.

This means that surds may need to be changed into an equivalent form, just like when we add fractions, and we want a common denominator.

Let's start with this question.

I need to write root 45 plus root 20 in the form k root 5.

Hmm.

The question's actually given me a big hint here.

It's telling me that they want the radicand to be 5.

Now I know that 5 is a factor of both 45 and 20.

So what I'm going to do is I'm going to start by rewriting both of those surds with 5 as one of the two factor pairs.

Square root 45 can be written as follows.

We know that 45 is the same as 9 times 5.

Ah, there we go.

We're simplifying again with those perfect square factors.

Root 45 is equivalent to 3 root 5.

And there's that root 5 I was expecting.

Root 20, well I can write the 20 as 4 times 5, which means that root 20 can become 2 root 5.

I can now write out that statement again only using the simplified forms. In other words, root 45 add root 20 is equivalent to 3 root 5 add 2 root 5, or 5 root 5.

And I can check.

Is that in the required form? Yes, it is.

There's a coefficient and there's the root 5.

Just like I was asked to do.

It's now your turn.

Please write root 75 add root 48 subtract 2 root 12 in the form k root 3.

Pause the video while you do this now.

Welcome back.

Did you start by simplifying? Remember the fact that you want root 3 to be part of the answer giving you a big hint into how to simplify these surds.

Root 75 is the same as the square root of 25 multiplied by 3, or 5 root 3.

48 can be written as 16 times 3, so we simplify to 4 root 3.

And then the 2 root 12, well, 12 can be simplified into 4 times 3, so that becomes 2 root 3, but they're already multiplying by 2, so therefore 4 root 3.

We can now write this statement out again using the simplified forms. As you can see, our expression now becomes 5 root 3 add 4 root 3 subtract 4 root 3.

So 5 root 3 is our answer.

And we check.

Is that in the required form? Yes, it is.

It's now time for your final task.

There are three parts to this.

In part a, we'd like to complete the empty boxes in the pyramid using the bricks.

Each box is the sum of the two boxes directly below it, i.

e.

, the two boxes it sits on.

Use the bricks to complete this.

Pause the video while you do this now.

Welcome back.

Let's look at the other parts of the task.

Now in part b, there are two sub parts to this.

We'd like you to calculate the perimeter of each of the two shapes you can see.

You need to leave your answers in exact form.

In other words, no decimals here please.

'Cause I can see we have surds so I know that I have irrational numbers.

These are going to be non-terminating decimals.

So when we say leave your answer in exact form, we mean leave it with surd notation, but it should be fully simplified.

So for each shape, please calculate the perimeter.

Pause the video and do this now.

Welcome back.

Let's go through the solutions.

Let's start with our pyramid.

Now you can see on the screen I've shaded where the bricks went that were around the pyramid so you can see which ones were there before and which ones we've added in.

Remember, summing the two bricks below gave you the value of the brick sitting on top.

If we consider just the top three bricks for now, root 12 subtract root 18 add root 18 subtract root 3, while the negative root 18 and the positive root 18 will cancel each other out, so I'd be left with root 12 subtract root 3.

But root 12 can be written as 2 root 3, so I'd actually have 2 root 3 minus root 3 giving me the root 3 for the top brick.

So I know that one works.

The same reasoning can be used to determine the other four bricks.

It might be wise to fully simplify because that way it's very easy to sum the bricks because we'll have like terms. In b, we asked you to calculate the perimeter of two different shapes.

Let's start with this rectangle.

We know that in order to calculate the perimeter of a rectangle, you have 2 lots of the length and 2 lots of the width.

What I've done here to be very clear is I've written out each length twice.

So we have 2 add root 18 plus 2 add root 18 plus 3 subtract root 2 plus 3 subtract root 2.

And I then gather the like terms. So 2 add 2 add 3 and 3 gives us 10.

There are 2 lots of root 18.

And then I've got negative root 2 take away another root 2, which gives us a total of subtracting 2 root 2.

Now is that fully simplified? Well, the root 2 definitely is, but the square root of 18 isn't because 18 has a square factor of 9.

So I know I can simplify that surd, leaving us with a final result of 10 add 4 root 2.

Let's look at our square now.

Well, we know that a square has four sides of equal length and we can see that from the notation on the screen.

So what I've done is I've written out one of the lengths, root 20 subtract 2 root 3, four times, and summed them.

In total, this will mean I have 4 lots of root 20 subtract 8 lots of root 3.

Now I haven't stopped there though because the root of 20 can be simplified, and that gives me my final expression, 8 root 5 subtract 8 root 3.

It's now time to sum up what we've done today.

Like surds can be added in the same way as like terms in algebra.

And sometimes it is helpful to simplify before we complete the addition because we require our surds to be like terms. Well done.

You've put in lots of effort today and done really well.

I look forward to seeing you for our next lesson.