Lesson video

Hi, welcome to today's lesson on rationalising a single term denominator.

By the end of today's lesson, you'll be able to use the technique of rationalising the denominator to transform a fraction to an equivalent fraction.

Our lesson has two parts today, and we're going to start with part one on the simplest form.

A fraction is in its simplest form when it can be written a over b where a and b are integers with no common factors, except one, of course, but one is a trivial factor.

In other words, dividing by one isn't going to change what's there.

Let's consider these two fractions.

Are either of these in the simplest form? Well, four over 1.

5 isn't because 1.

5 is not an integer.

Three over 22, of course, is in its simplest form.

both a and b are integers and three, which is the only factor of the numerator that isn't one, is not a factor of 22.

So in other words, no common factors.

Now, similar to fractions, the radicand should always be written as an integer.

Now, this may mean that you need to convert a mixed number to an improper fraction so that you can simplify.

Let's have a look at what I mean.

We're going to simplify the square root of seven and 1/9.

Now, initially, I'm not sure how to do this, so I'm going to convert to an improper fraction first.

This leads me to 64 over nine.

Now, I looked at my rules for dividing with surds, and I know that when I square root this fraction, I can write an equivalent calculation, which is the square root of the numerator divided by the square root of the denominator, or in other words, root 64 divided by root nine.

Well, that's a lot easier.

The square root of 64 is eight and the square root of nine is three.

In other words, that surd simplified is eight over three or I could say two and 2/3 if I wish to convert it back to a mixed number.

Well, that suddenly got a lot easier.

Breathing a bit of a sigh of relief now.

Now, of course, I may not have got to a rational answer.

There was a chance there'd still be a surd there but it is a lot simpler than it was before.

Now it's your turn.

Try simplifying the square root of 30 and 1/4.

Consider whether or not going to an improper fraction may be the best route forward.

Pause the video now while you have a go.

Welcome back.

Let's see how you got on.

Well, if you changed it to an improper fraction, you would've had 121 over four, which we can then break up into the root of 121 divided by the root of four.

Well, square root of 121 is 11, and the square root of four is two, which is 11 two or five and 1/2.

Oh, it is nice when we get to a rational answer.

It just looks so neat and tidy.

Remember though, that may not always be the case.

For your first task today, I'd like you please to simplify each surd.

Leave your answer as an improper fraction.

In other words, you don't need to convert it back to being a mixed number.

Pause the video now while you complete this task.

Welcome back.

Let's see how you got on.

For a, I converted it again to an improper fraction.

So 25 over 16, which simplified to five over four.

For b, I had one and 11/16, which became 27 over 16.

Well, the 16 at least simplified.

Square root of 16, after all, is four.

The square root of 27 I couldn't simplify to a rational number, but I could at least simplify the surd to become three root three.

In c, my square root is six and 1/4, so I converted to an improper fraction, which gave me 25 over four.

Square root of 25 is five, square root of four is two, so my answer is five over two.

In d, again, I converted to an improper fraction, so I had 27 over four.

Well, the square root of four is two.

That's my denominator sorted.

And the numerator becomes three root three.

It's now time for the second part of our lesson today and this is where we're going to be looking at a single term denominator and this process that we call rationalising.

We're gonna start by considering the two calculations we just did as part of our task.

What is the same and what is different about the two simplified surds? What did you come up with? Here are some things that you might've said.

You said both simplified surds have a surd as part of their numerator, and they do.

They both have the root three.

You might have said that neither of them has a surd in the denominator.

You might have said that the numerators are exactly the same but the denominators are different.

In fact, both of these simplified surds are, in fact, fully simplified and we'll see why.

Both of our simplified surds resulted in fractions where a surd was present in the numerator but not in the denominator.

A simplified fraction should not contain a surd in the denominator, and that's convention.

What happens though if there was one there? Well, in order to simplify, we'll need to write an equivalent fraction.

How could we do this though? The process is called rationalising and it's where we remove the radicals from an expression.

In fractions, the convention is to rationalise the denominator.

I.

e.

remove the radical sign from the denominator and we do this through equivalent fractions.

The two fractions you can see here are actually equivalent to each other, but the one on the left is not simplified because there is a radical sign in the denominator.

The fraction on the right, however, is fully simplified.

Although there is a radical sign, it's in the numerator and this is acceptable.

It's just the denominator we don't want to see a square root sign in.

Let's look at how we would do this.

We're going to rationalise two divided by the root of three.

What could I multiply root three by to remove the radical sign? That's right, I can multiply it by itself.

I can square it.

Root three multiplied by root three is three.

Now, in order to maintain equivalence, I need to multiply both numerator and the denominator, not just the denominator.

So because I'm multiplying the denominator by root three, I do the same to the numerator, so that I formed an equivalent fraction.

I have scaled my fraction by the same amount.

Two multiplied by root three is two root three, so in other words, two divided by the square root of three is the same as two root three divided by three.

This is now a fully simplified fraction because my denominator has been rationalised.

When I look at both numerator and denominator, each term is written as simply as possible.

It's now your turn.

I'd like you to rationalise two divided by the square root of two.

In other words, write an equivalent fraction where there is no radical sign in the denominator.

Pause the video and do this now, please.

Welcome back.

How did you get on? You should have multiplied by root two over root two.

When we do that, the numerator becomes two root two and the denominator becomes two.

Now, you might have stopped there, except I can divide both numerator and denominator by two, which I should do so that I fully simplified my fraction, leaving me with a result of the square root of two.

What would you multiply by to rationalise the denominator for this fraction? You have three options here.

Pause the video and make your choice.

Welcome back.

Which one did you go for? You should have picked B.

Remember, we know we want to multiply so the denominator becomes rational.

Root three times root three gives me three, so that will work.

However, I want an equivalent fraction, which means that I would've multiplied this fraction by one.

One can be written as any number divided by itself but that does mean I need my numerator and my denominator to be the same.

In other words, root three over root three.

In case you did do it, if you had rationalised that, you would've ended up with four root three over three.

What about this fraction? I'd like you to rationalise the denominator of one plus root three all over root two.

Which one would you go for? A, B, or C? Pause the video now and make your choice.

Welcome back.

Which one did you go for? It should indeed have been A.

The denominator is root two, so we need to multiply it by itself.

In order to make sure we have an equivalent fraction, we have multiplied both numerator and denominator by root two.

What about where we have a fraction where the surd in our denominator has a coefficient? Should I multiply by two root three or perhaps by something else? Exactly.

I don't need to multiply by everything I see on the denominator.

That's a trap that we might fall into if we weren't really thinking hard.

I just need to rationalise the surd.

In other words, two multiplied by root three multiplied by root three again will mean I'll get to two times three, which is six.

If I'd multiplied by two root three, I would've ended up with four multiplied by three, which would be 12 instead of six.

Actually, what happens if I'd multiplied by two root three, my denominator would've been 12 and I would've had a larger numerator.

I'd have ended up having to divide both numerator and denominator by two at the end.

In other words, I'm giving myself more work to do.

I just need to rationalise the surd.

So this gives me three root three subtract three, all over six.

If you had multiplied both by two root three, then we would've ended up with six root three minus six over 12.

I had to have divided everything by two.

As it is, I've still got to divide here because that still wasn't as simple as it could be but it's not quite as much work.

Here I can see that all my terms had a common factor of three, so I've divided each turn by three.

It's now your turn.

Rationalise the denominator for the fraction on the right.

Pause the video and do this now.

Welcome back.

How did you get on? What should you multiply by? That's right.

Root two.

Both numerator and denominator should be multiplied by the square root of two, meaning that my numerator becomes root two plus two and my denominator becomes six.

What would you multiply by to rationalise this denominator? Pause the video and make your choice.

Did you choose C? Remember, we only need to rationalise the surd.

What do you multiply root five by to achieve a rational value? That's right, by itself, so root five over root five will rationalise the denominator.

You'd actually end up with five root five over 15.

Now, the five root five over 15 would then simplify because both numerator and denominator have a factor of five, so you'd end up with root five over three.

Now, what would you multiply by to rationalise this denominator? Pause the video and make your choice.

Welcome back.

Did you pick C? Well done if you did.

A would work, but remember, it's not the most efficient method.

We don't need to multiply by that coefficient as well.

It's now time for our second and final task.

For a, b, c and d, I'd like you please to rationalise the denominator in each case.

Pause the video and do this now.

Welcome back.

It's now time for the final two parts of our task.

In e, I'd like you to simplify and in f, something slightly different.

I'd like you to complete the number line that you see.

Pause the video and do this now.

Welcome back.

Let's go through our solution, shall we? We're going to rationalise the denominators.

We multiply a by root two over root two, giving us an answer of root two over two.

In b, we multiply by root five, so we end up with 15 root five over five or three root five.

In c, we're going to multiply by root two to be as efficient as possible, so root two over root two is the same as five root 12 over four.

Only, we can simplify the square root of 12.

It becomes two root three.

Don't forget, we've already got a coefficient of five, so five times two is 10.

10 root three over four, but we can divide both the 10 and the four by two, so that leaves us with five root three over two.

Indeed the first thing I'm gonna do is I'm going to write that as a fraction.

Now I'm going to go about rationalising the denominator by multiplying both numerator and denominator by root two.

This gives me three root 20 over eight and then I can simplify the root 20 so that my numerator becomes six root five.

I can then take out a factor of two from both numerator and denominator, giving me three root five over four.

In part e, I asked you to simplify.

First thing we're gonna do is turn the radicand into an improper fraction so it becomes 64 over five.

Well, the square root of 64 is eight but the denominator stays as root five.

I need to rationalise that denominator because it's not currently in the simplest form and this results in eight root five over five.

Now, to complete the number line, it may have seemed a little unusual at first.

You may have thought to yourself it's just gonna be some multiples of two maybe.

What I'm going to do though is fully simplify the surds that I do have.

The square root of eight can be written as two root two and the square root of 32 can be written as four root two, and when I do that, I can work out the scale of my number line, meaning the two missing values should be three root two and five root two.

It's now time to summarise what we've done in our lesson today on rationalising a single term denominator.

Rationalising is the process of removing radicals from an expression.

To rationalise the denominator, we use equivalent fractions to make the denominator rational.

In general, answers should be given with the denominator rationalised as this makes further working easier and it is convention to do so.

Well done today.

You've worked really well and I can tell you've put in lots of effort.

I look forward to seeing you in our next lesson where we're going to continue our work on rationalising denominators.