Lesson video

Hello, everyone.

Today's lesson is on rationalising surds of a more complex nature.

Hi, everyone.

We're going to start off by talking about conjugate pairs.

What's the same and what's different between these two expressions? They are identical.

But hopefully you can see one has a positive sign and one has a negative sign.

These expressions can be known as conjugate pairs.

So do you see a conjugate pair here? Well, no.

If we look here, we have a negative and a negative sign.

They're identical expressions.

They're not conjugate pairs.

What about this example? Conjugate pairs? Yes or no? Yes.

These are conjugate pairs.

The expressions look exactly the same.

One has a negative sign, one has a positive sign.

Multiplying conjugate pairs.

Here's an expression.

Here's its conjugate.

If I multiplied these two together, something very interesting happens.

Do you see? I end up with a number which has no surd in it.

It is a rational number.

I can show, using a multiplication grid, why this happens.

I'll collect all my like terms. And there we go.

Do you notice? Here, these two terms equals zero.

That just leaves me with two minus one, which gives me an answer of one.

So here's a top tip for you when multiplying conjugate pairs.

It's really easy to find the solutions, if I'm expanding these double brackets, simply by multiplying the surds.

And there we go.

And then multiplying those integers together.

And that just means that we've worked out the important parts of our equation.

Two minus one equals one.

Think about why that works.

So let's practise multiplying conjugate pairs.

Pause the video.

When you're finished, return.

Here are the solutions to question number one.

What do you notice about all the answers? Yes.

They're all integers.

They are rational numbers.

We're going to use our skill of multiplying conjugate pairs to rationalise a fraction with a surd.

Let's take this one step at a time.

First of all, find the conjugate.

If I multiply the numerators and the denominators together, here's what I end up with.

Here's the answer I obtain.

We need to simplify it.

I can use the grid method to simplify the denominator.

That will give me an answer of one.

Anything divided by one will just find you the number itself.

And so the final answer is at the bottom there.

So there's a lot of steps involved here.

This is not an easy task.

Take one step at a time.

Find the conjugate, multiply the numerator and denominator by the conjugate, and then simplify your answers.

So let's have a look at one more example.

Let's give it one more go.

Rationalise this fraction.

Find the conjugates.

Multiply the numerators and denominators together.

So let's simplify the answer we obtained.

We can use the grid method to help us find the denominator.

And you should have had a denominator of two.

And then on top, a square three minus one.

There you go.

So here are some questions for you to try.

Take your time with them and come back once you've had a go at a few of the questions.

There are three slides, but do as much as you feel you can manage.

Here is solutions to question two and three.

How did you do? Question three, hopefully you noticed that question d has been simplified to two minus the square root of two.

If you've written it as two minus the square root of two all over one, that's fine.

But we want to simplify everything.

Here are the solutions to questions number four and five.

You would have needed a calculator to work out which was the largest number, but if you managed without one, fantastic.

Here are the solutions to questions number six and seven.

Amir made the mistake of forgetting to change the signs.

And question seven was, although on the surface, it looks like a fairly quick question, required a lot of working out.

And remember that.

This is a tricky skill.

You take your time with it.

If you didn't get it to question seven, maybe come back to it at a different time.

Well done.