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Hello and welcome on this lesson on Fractional indices 2, with me Miss Oreyomi.

For today's lesson, you've been needing a paper and a pen or something you could write on and with also it would help if you minimise the distraction by putting your phone on silence and trying to get into a space with less noise and less distraction.

So if you need to do those things now before the lesson then please do so and then press play to resume with the lesson.

Okay for your child this task, you are being asked to think about the six numbers on your screen.

What is the same and what is different about these numbers? Try to think for a hint.

Think about the numbers eight and 64.

What is different? And what is the same about these numbers? Pause the screen now, take a moment or two to think about this and then once you're ready press play to resume with the video.

Okay, hopefully you saw that this here is the same as that in the sense that I could write this cube root of eight as eight one third times eight one third.

Hopefully you also saw that this, because we've got powers we're more to playing out the powers.

So I could multiply two times one third I get two over three, one third times two, I get two thirds.

So it is essentially the same because it is commutative law.

And then for this one, hopefully you are able to see that this and this were the same in the sense of how they're written, but also did you notice that if you workout every single one of this if you evaluated every single one of this you will get the same answer.

And let's do that now.

One eight third times eight third so that is the cube root of eight which is two times and over cube root of eight so that gives me four.

Eight squared is 64 and the cube root of 64 is four.

Cube root of 64 is four, eight squared is 64, so again, the cube root of 64 is four.

This is essentially the same as that as we mentioned before.

So that is two times two, which is four this one it's going find the cube root of eight, two then square your answer, which is four.

Again find the cube root of eight, two square your answer four.

So how come these are all the same? So we've established from here that 64 is a cube number and eight is a cubed number.

So what is happening here? We've got eight, two third eight raised to the power of two thirds.

If you remember from last lesson, our numerator if the lesson fraction indices one our numerator was never greater than one but in this case, we've got a numerator that is bigger than one.

Essentially, when I have a base that is raised to a numerator and denominator, fractional fractional number I could write this as my base is put inside my root.

I then put my denominator outside my root and raise it all to the power of my numerator.

If that is confusing for you, fair not we are going to do several examples.

So I've got eight and that is raised to the power two thirds, wait, I'm going to start by writing my base of eight.

I am then going to, 'cause this is the same as raising eight to the power of two thirds.

So it's going to be the cube root of eight multiply that answer or rather square that answer.

So that is that squared over there.

That is the I'm going to change the colour so you could see that this is the denominator over here and the base which is eight.

So my base, which is eight over here.

And as we worked out for my task the cube root of eight is two square my answer that is going to give me four.

Let's do another example, 25 raised over five thirds so first thing I'm going to do I am going to put my base inside my root symbol.

what's going to go besides my square root symbol, it's going to be three 'cause this denominator means find the cube root of my base 27.

So I'm going to put three here and what am I going to raise all of this to? Yes, I am going to raise all of this to five.

So let's work this out.

What is the cube root of 27? It is three, right? So that means three to the power of five.

And that is 243.

Let's do one more example.

I've got 256 raised to three quarters.

Okay, I want to write this out in a way that I can understand.

I am starting with 256, that is my base number.

I am going to put my roots symbol.

What is going to go next to my root? Well, it's asking me for the fourth root the quarter of the fourth root of 256.

So I'm going to put four here and then I am going to raise whatever on so I get to the power of three.

So if I'm looking for the fourth root of 256 just using my calculator over here that says four.

Four to the power of three is 64.

So essentially when your base is raised to a fractional power and your numerator here is greater than one this is how you go about working it out.

You put your denominator besides your square root over here and then you raise it to your numerator.

What if they've given you this and they're telling you to write this as an index form? Well, let's do the first example.

I always start with my base.

What is my base in this question? It is j isn't it.

So I'm going to write j over here.

What is my denominator? My denominator is always the number here next to my square root.

So in this case is going to be three.

And what am I raising the whole thing to? Two, so its going to be j raised to the power of two third and I write it like so.

Let's do the second example.

What is my base in this question? Nine, what is my denominator? It's going to be four and my numerator is therefore five.

So if you have a question and it says write an index for me, it means you need a base raised of the power of something, your turn then.

You've got five questions on your screen.

Pause the screen, a pause your video now attempt all five questions and then press play.

And we'll go for the answers together.

Okay, hopefully you had a go of that and were able to solve it.

Okay, I've got 16 raised to the power of three over two first thing I'm going to do.

I am going to write 16 as my base.

I don't need to put a two there because this is a square root symbol but I could put a two if I want to.

You don't have to, but I could write a two there.

And then I'm going to raise everything to the power of three.

So what is the square root of 16, four four raised to the power of three is 64.

Next one.

Again, I don't need to put my two over here because that is a square root symbol but I'm going to put my four as my base and I'm going to raise my answer to the power of five.

So that gives me 32.

Next one then, same as before I've got my four I've got my two and I'm going to raise my answer to the power of seven and two to the power of seven is 128.

Let's write these then in index form.

What am I starting with? Starting with my bases r isn't it? So I'm going to write r over there and what am I raising it to? Well, my numerator is four.

The number next to my base over here and my denominator is five.

Next one, start with my base my base in this one t what is my numerator? The number next to my base, which is eight.

And my denominator number next to my square root, which is six.

So I heard that makes sense because it is now time for your independent task.

You are to attempt every question on your sheet and then resume watching the video and we'll go for the answers together.

So pause your screen now and go through the questions and press play, and we'll go for the answers together.

Okay hopefully you found that challenging and engaging and interesting when are going to go over the answers.

If our first one we have a 1000 raised to the power of two thirds.

So I am going to put the three over there put a 1000 in my roots symbol and raised everything to the power of two.

The cube root of a 1000 is 10, 10 squared is 100.

Next one then, I have this raised of the power of five so my answer is 32.

Next one.

I'm looking for the sixth root of 64.

So a number that I can times by itself six times to give me 64.

And that number is two.

I am then going to raise my two to the power of five.

So again, my answer is 32.

This one then I've got 625 and I'm looking for the fourth root of 625.

I believe that is five.

And then five to raised to the power of five is 3,125 so the fourth root of 625 is a five.

And then I'm raising five by itself, five times.

And I get 3,125.

Next one then, I want to write each of the following in its index form.

So I've got w always my base.

The three is always my numerator the number next to my square root is always my denominator.

So I'm going to have my three over there and I'm going to raise it.

The two is my numerator.

So two goes there.

Hey, I've got v.

Then I've got three over seven here.

I've got k.

And then I have four over three and then I've got t and I've got five over six.

Write each of the following in the form 64 raised to the power of something.

So 64 race of the power of something would give me eight.

What is that number going to be is going to be 64 raised to a half, because this is the same as writing that the square root of 64 is eight.

What of this one? This is going to be the same as 64 raised to a third, because a third the cube root of 64 is four.

what of this one.

If I know that the cube root of 64 is four, what do I need to times that by what do I need to square my answer by to get what, on two times my answer by to get 16 after times a by itself, isn't it.

So I have to square that number.

So it's going to be 64 raised to the two thirds because I've got the cube root of 64 is four.

And then I square my answer four times four gives me 16.

Onto your explore task then using the numbers two to seven how many ways can you make the statement true.

I have tried mine on this notepad so I we're going to go for this one example that I came up with.

So for the first one, I wrote two here and I've got three there.

So essentially a square root of 64 is eight and eight to the power of three.

I got 512 here.

So how many ways can you make this statement true means how many ways can you this number less than this number and that number less than that number.

So for this one, I've got six and five, say 64 to the power of six divided by 64 to the power of five.

And I got 64 there.

And then the last one, I had two thirds because the cube root of 64 is four and four squared is 16.

So this is one way of making this statement true.

How many other ways using the numbers only between two and six.

How many ways can you make this statement true? Pause your video now and attempt this and see how many ways you can come up with.

We have now reached the end of today's lesson, a very big well done for sticking all the way through and completed all your task before you go, make sure you complete your quiz just to show yourself what you've learnt and you can even show off your score to your family and friends as well.

So again, very big well done and I will see you at the next lesson.

Bye.