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Hello, and welcome to this lesson on two to the power of n that's part of the growth and decay series.

I'm Mr. Thomas, and I couldn't be happier to be teaching you yet again for your lesson for today.

So make sure you've got your calculator at ready.

Make sure you've got a pen and paper you can write things down on.

I mean, writing on my calculator, how weird? And make sure you're ready to go and that you're in that quiet space that is distraction-free.

So without further ado, let's get onto the lesson.

So Xavier is putting some rice on a chessboard.

He puts one grain on the first square, two grains on the second square, four grains on the third square, continues to doubling it, doubling it, doubling it.

So I'd like you to have a think about the total number of grains of rice once Xavier finishes the whole board like that.

So you're going to go all the way across and fill up all those spaces, doubling it each time.

So have a think about it.

I'm going to give you a minute to do that, but just have a think about it.

Off you go.

Great.

So let's go through it then.

So guess the total number of grains of rice once Xavier finishes the whole board like that.

Well, what we're going to have is this would be, it would be two to the power of zero there, wouldn't it? 'Cause that would be one.

Two to the power of one there, so that would be two, of course.

Two to the power of two there, which of course is is four.

You noticing the pattern now? So we go three, four, five, six, seven across, and then eight, nine, 10, 11, 12, 13, 14, 15.

So this is seven.

15.

We're going to add on seven each time.

Aren't we? So we're going to get to 22, 29, and then 36 down here.

And then, what do we need to do? Add on seven each time, right? So 43, 50, and then 57.

Goodness me.

So that's going to be two to the power of 57.

Now, if you type that into your calculator, you're going to get a very, very big answer.

You're going to get 1.

441151881 x 10 to the power of 17.

So that's going to be, this.

What do we get? We get, going to check it as I go through.

I'm going to get, so we have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17.

If I put my commas in there, easier for me to see.

So approximately it's going to be, that is a thousand.

That is a million.

That is a billion.

That is a trillion.

That's going to be 144 quintillion, right? That is, that's astronomical.

I can't begin to think how big that is.

That is just insane.

So really, really big number there.

Right? So big.

You probably get some way under that if I'm really honest.

Can't imagine you've got that.

I didn't even guess that.

So have a go and think about how of a number that is.

So difficult to think of.

So this whole idea is that, Xavier saying this pattern is growing by the same amount each time.

I want to explore that by considering what he said there because it is growing by the same amount in some ways, right? It's multiplying two, but it's not a constant amount.

It's not constant amount.

And that's really important to be aware of.

It relies on the previous number.

It relies on that previous number.

And what ends up happening is something like this.

Is that you can draw a curve and it goes exponentially upwards.

And it hits me right at the top here.

You can see it hitting me, right? It keeps going up and up.

It would go through my roof.

It would go right up to this sky.

Really, really high up so quickly, right? It starts off a very small amount and grows and grows as time goes on.

Relies on that prev, doubles each time it's much quicker than adding on say 10 each time.

Doubles in time.

So powerful.

So what I want us to think about is this independent task here, whereby we've got y = 5 to the power of n.

I want you to fill in those blanks there with what that would be, plot the graphs and see what you notice, what similarities and what differences are there, and then think about those integer values there.

So I'm going to give you 15 minutes.

It could take quite a while I think to do this.

So pause video now and have a go at that please.

Off you go.

Excellent, let's get back to it then.

So for this one, we've got, remember anything's the power of zero is one, then to the power of one would be five 25, 125, 625.

But remember with this one, it's going to be 1/5, 1/25, 'cause that is of course five squared, isn't it? So we're doing the reciprocal remember, yeah, in previous videos.

What about this one? This would be 1/10 and then 1/100 as a result of that.

So mark it up right or wrong if you've got that.

So plot the graphs.

What similarities do you notice? So, if I do a very quick sketch of this, what this is going to look like is that it's going to look like that.

It's going to cross over on the y-axis at one.

It's going to be exponential.

Same for this one, but it's just slightly steeper is that it looks like this.

So that's y = 10 to the power of n, and that is y = 5 to the power of n.

Still crosses over though really crucially on the x-axis.

So that's one of the similarities there are.

And differences, it's just more exponential.

So for the consecutive integer values of n and m, we can think, well, that would be three 'cause three to the power of three would give us 27.

And then three to the power of m would be four there.

So it could be five or six or seven or eight or whatever number you wanted, but it has to be consecutive.

So be very, very careful.

Consecutive meaning the next one along.

This one here, you're going to have three and four again, aren't you? 'Cause 125 would be five to power three and then five to power four would be 625, so 200 lies between them.

Very good if you managed to get that without me helping.

So for your explore task, now, I want us to consider some apply to biology, here, and it's to do with cells dividing in two to create two new cells, and it occurs every minute.

So how long after would be, there'll be more than 10 cells? 1,000 cells? And then 10,000 cells? So pause the video now and have it go for the next, I'm going to say, eight minutes.

Off you go.

Very good.

Let's go through it then.

So we can think about this as a problem, right? If we know that it creates, if we start off, right, with one, one thing here, and then every minute, so one minute passes, we get another two cells.

They then split off into two more, et cetera, et cetera, et cetera.

Do you see that this could be two to the power of zero? That's one.

Two to the power of two, so one, is equal to two.

Two to the power of two is equal to four.

Et cetera.

So those refers to the minutes, n being the number of minutes.

So what we can do from there is we can say, well, how many more minutes would it be until 10 or more cells? Well, we get two to the power of three would be eight, and then two to the power four would be 16.

So it'd be four minutes.

What about 100 then? Well, you can play around with your calculator at this point.

But two to the power of, two to the power of seven would allow for 128, wouldn't it? And two to the power of six, of course, is 64.

So I can then say, I can then say that it would be seven minutes, then.

What about 1,000 cells? Well, we can keep growing up and up and up, and what we see is that two to the power of nine would be equal to 512, and then two to the power of 10 would be equal to, what would it be? 1,024.

So we can see this one will be 10 minutes.

And then the final one for 10,000 cells.

Well, that's interesting.

If we play around with our calculator even further, what we get to is two to the power of 13 would be 8,000, there we go, 8,192, and then two to the power of 14 would be 16,384.

So we can see this one would be after 14 minutes.

So we can see how quickly cells can reproduce and how that applies to that exponential model of y = 2 to the power of n, which is what this was focusing on.

Really amazing.

If you keep up with that.

Really good job, well done.

So that brings us to the end of the lesson.

Really, really big congratulations if you've managed to keep up, 'cause that's some quite complex topics, there.

And it's really applied in some ways, which is really nice to see.

You can see it with investments, you can see it with biology, et cetera.

Cells reproducing, investments growing over time, all sorts.

So it's really important that you can keep up with that and apply it to real-life context.

So this is our penultimate video.

I hope to see you in that final episode, so we can smash out the park and really cover exponentials in a lot of depth.

So, for now, take care, and I'll be seeing you.

Goodbye.