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Hello, and welcome to this lesson on indices and radicals with me, Miss Oreyomi.

For today's lesson, you'll be needing your paper and your pen.

Also, if you could minimise the distractions by putting your phone on silence and trying to get into a space where there's less distraction.

So if you need to pause the video to do that now, then please do so and press resume to carry on with the lesson.

So you try this task, you have five numbers on your screen and your job is to sort them in order from smallest to largest.

So pause the video now and attempt this task, and then once you're done, press resume to continue with the lesson.

Hopefully you had a go at that.

We know that numbers like this is called a surd.

And we know that surd is an expression that has a square root or cube root or any type of root symbol.

And we know that numbers like this is called indices.

So that means the number has been multiplied by itself.

So let's evaluate each number then.

Two cubed we know is eight, square root of 25 is five, cube root of 64 is four, negative six Squared is 36, and negative three squared is nine.

So our smallest number would be three root 64, followed by root 25, followed by two cubed, followed by negative three all squared, Followed by negative six all squared.

So check in your work, making sure you've got the right answer.

Hopefully on your screen right now is nothing new.

Hopefully you've seen this before, just to recap, when we're told to simplify, when we're told to simplify surds, it can only be simplified if the number in our roots has a square number as a factor.

So if I'm looking to simplify root 18, how can I break it into, either, one of its numbers that I multiply together is a square number.

Well, nine is a square number, and if I do root nine times roots two, multiply these two together, that would still give me root 18.

But I can simplify this.

What's root nine? Root nine is three, and root two will remain in the squared form because two is not a square number.

Let's move on to this one then.

I want to simplify root 12 times root nine.

I could write root 12 as root four times route three, can't I? And then multiply that by root nine, which is three.

But I'm not quite done because, what's the root of four? This is going to be two times three, times root three, so my answer is going to be six root three.

What about 63? How can I write this then so that one of my factor is a square number? Hopefully you've said root nine times root seven.

Root nine is square of nine, is three.

And I'm going to leave root seven in its surd form.

So my answer is going to be three root seven.

Let's look at the last example then.

Root 63 divided by root seven.

I could write this as root 63, divided by root seven, can't I? Which is going to give me root nine because 63 divided by seven is nine.

So my final answer is going to be, my final answer is going to be three.

Let's tackle a couple of more examples.

If I am adding surds, I can only add surds where the numbers in the root are the same.

So for the first one, would I be able to simplify this expression on my screen for the first one? The answer is no.

Because I've got root four here and I've got root seven.

They are not the same, so I cannot simplify it.

So it's literally just going to be the same, five root four plus four root seven.

What of this one? Can I simplify this any further? 14 plus five is 19, and then I am just going to write root three over there.

Let's look at c then.

I've got root 48, subtract root 27.

What is the highest squared number that would be a factor of 48? It is 16.

So I am going to write root 16 times three because 16 times three is 48 and subtract that from nine times three.

So that's an important point, actually, when I am trying to simplify, I am looking for the highest square number that is a factor of the number I'm trying to simplify.

So root of 16 is four.

And then my root three is going to remain as root three.

Again, root of nine is three and my root three is going to remain as a surd.

Four takeaway three is one.

So I am just going to write my answer as root three, 'cause that is the same as writing one root three.

Let's look at d over here, root of 20, I am going to write that as four times five, subtract, what is the cubed root of 64? That is four.

So four times five, I'm going to simplify that to two root five, and then subtract four.

And that is going to be my answer because I don't have a root five here to then simplify this any further.

Before you have a turn, before you have a go, let us work through some more examples.

I've got root seven all squared.

Now you may already know how to do this, but very quickly if I expand this out, so that will be root seven times root seven.

Seven times seven is? 49, so that would be root 49 which is the same as seven.

Root five times root 10.

I could simplify that by writing root 50.

Simplify that even further, I could say that will be root 25 times two.

Remember I'm always going for the highest square number that is a factor of 50.

And root 25 can be simplified to five and root two.

Last example then.

I've got a mixture of whole numbers and square root.

First thing is I'm going to put the whole numbers together.

So I'm going to do eight divided by two first, which is four.

And then I'm going to divide the square root.

So that will be root nine divided by root three.

Nine divided by three is three, my answer is going to be four root three.

Your turn then.

You're going to pause the video now, attempt all the independent task and then resume the video and we would go through the answers together.

I wonder how you got on.

I hope you are able to answer as many questions as you possibly can.

We are going to go through the answers together.

So the first one, the cube root of 125 is five.

And then I'm multiplying that by the fourth root of 81.

That means what number can I multiply by itself four times that would give me 81? And that number is three.

So my simplified answer should have been 15.

Let's look at the second one, root 40 plus root 40.

I can't write root 80 because root a plus root b is not the same as root a plus b.

However, root a times root b is the same as root a times b.

If you're wondering why, why not try with some numbers and see why we've come up with that rule? Going back here, so it's not root 80, 'cause I can't join them together, so it's going to be root four times 10, plus root four times 10.

So this is going to be root two root 10 plus two root 10.

And our final answers going to be four root 10.

I'm just going to draw an arrow that shows we're doing number three here.

Seven root three subtract five root three.

That is two root three.

For d, nine, remember we divide the whole numbers together first, that gives us three.

And then I would divide root 12 by four, that gives me root three.

So my final answer for d is three root three.

I know I am writing everywhere on the screen, but try to follow.

E, fourth root of 256 is four, subtract cube root of 64 is four.

So my simplified answer is zero.

Next one then.

Find the exact perimeter of the shape.

It's as a rectangle.

So the exact perimeter would be, me adding everything together.

So it would be three root three, plus another three root three, plus five root two, plus five root two.

So that answer, three root three plus three root three is going to be six root three, plus 10 root two So check in if you got that.

Let's think about number three.

We want to find the perimeter of the shape.

Root 10 plus root two, and then root 10 subtract root two.

I'm going to write this out.

So I've got root 10 plus root two, plus the bottom length, root 10 plus root two.

And then I am adding root 10 subtract root two, and doing that again for the other length, root 10 subtract root two.

Now, what can I cancel out to make this expression easier to work out? I've got positive root two would cancel out this negative root two.

And positive root two would cancel out negative root two.

How many root 10 do I have left? Four, so my answer is going to be four root 10.

Find the exact perimeter of the shape.

So again, I've got root 18 plus root two, plus root 18 plus root two.

I can simplify root 18 to it's factors by doing root nine times two, plus root nine times two again, plus root two plus root two.

This is going to give me three root two, plus three root two, plus root two, plus root two.

And in total, how many root two do I have? I have three, six, seven, eight.

So I've got eight root two.

For your explore task, you are to use the digits zero to nine, so numbers between zero and nine.

You are to then fill in these number frames, only this ones to, see if you can find a number that would sit in between A, B, C, or D.

So for example, if I write root 49 here, root 49 would be in between, would be at B because root 25 is five and two cubed is eight.

Root 49 is seven, so that is what I am going to put there.

So using numbers zero to nine, can you fill in these number frames so that each one was sit at A, B, C or D.

If you finish using the square root and cube root, can you try to using indices? Can you fill in the numbers either each number, using indices sit between A, B, C, D, or E? We have now reached the end of today's lesson and a very big well done for sticking all the way through and recapping your knowledge on surds.

So hopefully you already know all these things.

If you don't, then feel free to watch the video again and go over this.

Before you go though, do complete the quiz as that shows you what you should know or shows you what you've learned into these lessons.

So please do so before signing off and I'll see you at the next lesson.