Lesson video

Hi, welcome to today's lesson on identifying square factors to support simplifying surds.

By the end of today's lesson, you'll be able to identify the highest square factor of any term, including those written with algebra.

There are going to be some important words that we use today in our lesson, and you can see them on the screen now.

We have factor, perfect square, surd, radical and radicand.

Pause the video and write down what you think each of these words means.

Welcome back.

Let's see what you've put.

A factor is a term which exactly divides another term.

A perfect square is a number that is the second exponent of an integer.

We usually shorten this to square number, so examples could be 16, 25, 64.

A surd is an irrational number expressed as the root of a rational number, and that's what we looked at in the previous lesson.

The root sign is referred to as the radical symbol, and the radicand is the value inside the radical symbol.

We'll be referring to those words today.

It's now time for the first part of our lesson, and that is on identifying surds.

A surd, remember, is an irrational number expressed as the root of a rational number.

Each surd represents a unique value.

A surd is a way of writing an irrational number exactly.

For example, if I have a square with an area of 10, we know that each side of the square must be the same length, and that multiplying two of these sides together gives us that area.

Well, rather than writing out what the square root of 10 is, it's an irrational number, remember, writing the square root of 10, i.

e.

stopping one operation before the end, is a nice way of writing this so that we can be exact.

Here's another example of where we could be exact with a surd.

Do you recognise this triangle? That's right.

It's a right angle triangle where I've been given two of the sides and I'd want to know the length of the remaining side.

What am I going to use to do this? That's right, Pythagoras' theorem.

In fact, if you'd like, you can pause the video now and have a quick go at calculating what the length of x is going to be.

When you first looked at this, we always rounded our answers, but now we know that we can leave things in the form of a surd, we don't have to write it as a decimal and therefore round it.

We can leave it exact.

Here, we would reach root five because one squared plus two squared is equal to five, and remember we would then need to square root to find the length of the longest side.

If a rational radicand can be expressed as a perfect square, then we don't have a surd.

Do you remember this from last lesson? The square root of 49, for example, we can write 49 as seven squared, so the radicand of 49 has been rewritten in an equivalent form, and because we can write it as a perfect square, we know that this is not a surd, and this is because squaring and square rooting are known as inverse operations, i.

e.

one undo the other.

Which of the following are surds? Pause the video now while you make your selection.

Welcome back.

Let's see what you put.

That's right, A and C are surds.

B and C can actually be written in a different way.

Let's start with B.

Four lots of nine is 36, and the square root of 36 can be written as the square root of six squared, which gives us a result of six.

Now, D is slightly more tricky.

0.

0625 can be written as 1/16.

Now, 1/16 is equivalent to 1/4 squared, and therefore the result is 1/4.

You may not have spotted that one, and that's absolutely fine because of course you could use your calculator, and evaluating that would've told you that you reached 1/4 as your value, so you would've known D is rational.

It's now time for your first task.

Identify which of the 12 values below are surds.

What we'd like you then to do is write an equivalent expression for the numbers that are not surds, that does not involve the radical sign.

In other words, just like you did in your check for understanding a moment ago where you rewrote some of them to show they were rational, that's what you can do here.

So in other words, evaluate it.

Pause the video and do this now.

Welcome back.

Let's start by identifying the surds.

C, D, F, J and L are all surds.

Let's look at the ones that aren't and show how they can be written instead.

Well, a 1/4 is the same as a 1/2 squared, which leads to a 1/2, and the square root to 4/9 can be written as the square to 2/3 squared, which leads to 2/3.

For E, if we multiply two squared by three squared, well, that's the same as saying two times three all squared, and therefore we have a square root and a perfect square leading to two times three, which is six.

You could of course have evaluated it and seen that was four times nine, which made 36, and the square root of 36 is six.

That's pretty much what I did in G in that I actually evaluated that instead.

Well, two times five times 10 is 100.

100 is the same as 10 squared or the square root of 10 squared is 10.

And then in H, I did the same thing.

I worked out the product of the radicand, which gave me 900, and the square root of 900 is 30.

This just leaves us with I and K, and here's where we had some algebra.

Well, the square root of anything squared is just itself, so in other words, in I, we reach a, and for K, I have ab times ab.

Well, that's the same as ab all squared, and square rooting that just gets us back to ab.

It's now time for part two of our lesson, and in this we're going to start to look at what the simplest form of a surd might be.

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

For example, the square root of 32, that's 16.

That is a perfect square and it is a factor of 32.

So I know that the square root of 32 is not in its simplest form because it has a perfect square factor greater than one.

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

Now, how to convert this from what you see here, so not in the simplest form, to the simplest form we will look at soon, but for the moment let's just identify if surds are in their simplest form or not.

The square root of 24 is in its simplest form.

Do you think that's true or false? You then need to justify your answer.

So either you could tell me a perfect square factor of 24 or you could tell me what the factors of 24 are to prove that there are no square numbers, it's your choice.

Pause the video now while you make your selection.

Welcome back.

Which one did you go for? That's right, it's false.

Four is a square factor of 24, so the square of 24 is not in its simplest form.

What about this one? ab where the b is cubed.

True, false and justify your answer please.

Pause the video now.

Welcome back.

Which ones did you go for? Let's, right, it's false.

B squared is a factor of B cubed because we could write B cubed as B squared times B.

Therefore there's a square factor.

What about the square root of 123? Pause the video and make your choice.

Welcome back.

This one is true.

If we write out the factors greater than one of 123, we have three, 41 and 123.

They're not square numbers.

For this second task, I'd like you to either discuss or write down what you think you will need to know in order to simplify a surd.

In other words, if I want to know whether or not a surd can be simplified, what is it I need to know? Pause the video and either discuss or write this down now.

Welcome back.

What did you say or what did you write down? I've said that you're going to need to know the square factors if you want to simplify.

So if there are square factors you can simplify and you need to be able to identify them or simplifying is not going to happen.

It's now time for the final part.

And this is where we're going to practise identifying the highest square factor.

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

We saw that the square of 32 is not in its simplest form because we could write that, well, we could write the radicand of 32 as 16 times two.

The square 221, however, is in the simplest form because there were no square factors of 21.

So to simplify, the first step is identifying those square factors.

There are multiple ways to do this, and we're going to look at two of them.

The first is to list the factor pairs.

We're going to do this with the value of 32.

Well, we have one times 32, two times 16, four times eight.

Which of these are our square factors? Well, we can see that one, four and 16 are our square factors, and the highest square factor would be 16.

It's now your turn.

Using the same method as me, please find the factor pairs of 72 and then list the square factors.

Feel free to circle the one that's the highest.

Pause the video and do this now.

Welcome back.

Let's see how you got on.

We have one times 72, two times 36, three times 24, four times 18, six times 12 and eight times nine.

Which of those are square factors? Well, there's the one, the four, the nine and the 36, and the highest of these of course is 36.

Now, there's a second method I mentioned, which is writing the number as a product of its primes.

This is something we've done before so you may be very familiar, but we'll recap just to make sure.

I'm going to find the square factors of 72, which is the number you just did yourselves.

So we already know what we should see here, one, four, nine and 36.

Let's see how writing the number as a product of its primes helps us get to those square factors.

72 is the same as two cubed times three squared.

So how can I identify the square factors from having written it in this form? Two squared is clearly one of the square factors because I can write two cubed as two squared times two, and the same is true for three squared in that I can see a three squared right there.

And then of course I could write it as two squared times three squared.

These will definitely be square factors because I can rewrite that as two lots of three all squared.

Well, when I evaluate these, look at the numbers I get.

Obviously one, but that's what we call a trivial factor.

More importantly, two squared is four, three squared is nine and two squared times three squared is 36.

So I've reached the same square factors, I've just used a different method.

I'd now like you to find the square factors of the value 2,700 by writing it as a product of its primes.

Pause the video and do this now.

Welcome back.

Let's see how you got on.

We can write 2,700 as two squared multiplied by three cubed multiplied by five squared.

Now, this gives us lots of square factors.

You can see them listed below.

I then need to evaluate each of these to write down what the square factors are.

Remember, one is our trivial square factor so it belongs there, and the others come from our evaluation.

Two squared is four, three squared is nine, five squared is 25, two squared times three squared is 36.

Two squared times five squared is 100, three squared times five squared is 225.

And lastly, two squared times three squared times five squared is 900.

In other words, there are lots of square factors there, and the highest of course is 900.

You've now seen two methods for identifying the highest square factor.

You could either have written the factor pairs or written the number as a product of its primes.

Which method did you prefer? To simplify a fraction, you identify the highest common factor of both the numerator and the denominator.

This should be quite familiar, for example, to simplify the fraction 24/32.

We look for the highest common factor of both numerator and denominator, in this case eight.

By dividing both by eight, we reach the fraction 3/4, which we know is an equivalent fraction.

You could do this of course over multiple stages by using a different common factor, but it's a lot more work.

As you can see here each time I've taken out a factor of two, and I have got to the same answer, but there are much more steps involved.

It's therefore more efficient if we can identify the highest common factor.

The same concept applies here to surds.

You can simplify using any square factor.

It will just take more steps.

The most efficient way therefore is to identify the highest square factor.

Remember, when we saw the square factors of 32, they were one, four and 16, and the highest of course is 16.

The highest square factor of 72 is nine.

Is that true or false? And you need to justify your answer.

So if you think there is a higher square factor then you need to write down what it is.

Pause the video now and make your choice.

Welcome back.

What did you say? That's right, it's false.

36 is a square factor of 72 and it's higher than nine.

Did you remember that from when we did our examples earlier? Find for me the highest square factor of 66,150.

Now, I think I've been quite nice here because I've already written that number as a product of its primes, so you can use that to identify which of A, B or C is the highest square factor.

Pause the video and do this now.

Welcome back.

Which one did you go for? That's right, it's C.

In the top one, remember, in A, we've got three cubed.

That's not going to be a square factor.

There's a cube there, so that's gone.

In B, look, we've got two written, but that's not squared so that's not going to be a square factor either.

Therefore it must be C.

It's now time for your final task.

We're going to start with part A, and for each of the values you can see there, we'd like you to find the square factors of those values.

So for the column on the left I'd like you please to find the square factors and do this using both methods, finding factor pairs and by writing the numbers or values as products of primes.

Then please identify what the highest square factor is.

Pause the video and do this now.

Welcome back.

It's time for part B.

You need to identify one strength and one weakness of each of the two methods.

That's the factor pairs method and the products of primes.

So one thing that's good about them and one thing that's not so good where you evaluate each strategy to decide if one is always better or just better under certain circumstances.

Pause the video and do this now.

Welcome back.

Let's start by checking your table.

Up on the screen now you can see in the blue boxes where I filled in our answers.

You'll notice that for the bottom two rows, there were too many factor pairs to list here, and that's because we used algebra.

In other words, there were so many combinations, we just couldn't fit it all in.

Now, that's also true for row three where we had a value because 1,800 had a large number of factor pairs and we wouldn't have been able to fit them into the box.

Well done if you started working on those.

It's now time to look at the strengths and weaknesses you identified.

For the factor pairs, you might have written down that there's no need for extra calculations because the squares are identified.

You might have said it's quite easy to get to the highest perfect square quickly, and then you can stop.

You might of course have said for negatives that it can take a really long time for those large numbers, and as you saw in the bottom three rows we couldn't fit them all into our box.

It also also identifies one.

One is not helpful.

Yes, okay, it is a perfect square, but it's a trivial factor.

It's not gonna help with our simplifying, and it is very easy to make mistakes.

For the prime factor method, it is less time consuming for larger numbers, and the indices make it easy to identify the square numbers.

However, it's a long method for smaller numbers and you then have to do another calculation to find the square factors.

Let's summarise what we've done in our lesson today.

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

The first step to simplifying surds is to identify the highest square factor.

This can be done by spotting the factor, using factor pairs or by writing the term as a product of its prime factors.

In the next lesson, we're going to take this technique and apply it so that we can actually simplify our surds.

Well done.

Thank you for all your effort and I look forward to seeing you in our next lesson.