What is 0 to the power of 0?

Let's think back. Where? What was the first thing we started with, under indices,

power

s, exponents? What is the definition of index? If I wrote, you know, A to the

power

of M,

what

does that mean? It means A for A for A, how many times? M times right? Well. So let's write that down. It's A multiplied by A multiplied by A. I don't know how many times that is M, right, but it was one or five or 60,

what

ever, right, so I'll write that that's M multiplied. That's what m and the power of m rather, okay, so we took this idea and said look, we can build on this.

Good? If you have two numbers like this A to m and a to some other correct base, then this is what we write. And if you multiply them together correctly, you can rewrite it like this and count it. But that took forever. Instead, we write. What we call index law, right? It is equal to that base and what is the power? What do you do for a living? When you multiply, you add the indices, right? So it's the M plus N case, it's ringing bells. Yes ok? exactly the same way if I do a division like that.

Well, you subtract the indices, okay, you take the difference, so far so good and then one of the most important things we covered was you know if these two numbers are equal m and n just A to the power of m divided by A to the power of n, you get a to the power to 0, right? But then we said well, that should be 1, that should be one. That was the last "More", if you will, that we established, so there are 1, 2, 3 laws, so we raised this question, okay? We said that for any number the power of 0 is 1.

In the same way you know that I can put this on the side 0 to the power of any number that is 0, right? 0 squared, 0 cubed, 0 to the power of 100... 0 to the power of are you paying attention waiting for what? (student:) I've been waiting for you... I've been waiting for you. (teacher:) great. Thank you. (student:) You're welcome. So, yeah, have you asked us about zero to the power of zero? Ah, well, yeah, so what happens when you put them together, right? any number raised to 0 is 1, any 0 raised to any number is 0. So what happens when you put them together?

OKAY? now! Yeah, therein lies a question, now this is where we're going to do it, and you're going to need your calculators for this. In fact, I'm going to take mine out. We'll also need everyone's calculators, so I'm going to ask you to help me run some numbers, okay? So get your calculator ready. So here's the thing, right? If you're looking from this angle, if you're approaching from this guy, you'd say it should be 0. If you're coming from this direction, you'd expect 1, so is it one of those options or is it something like that? completely different?

Ok, this is what we are going to do. I'm going to make a table. You. You don't need to draw this table, but you will help me complete it, okay? hmm What I want us to think about is what do these numbers equal if I'm starting to get closer to... you know... really small numbers. so have really small numbers. So if you want, what we're going to do is something called boundaries. I will explain it shortly. But it seems like you know that, right? and this is what you need on your calculator. In the middle at the top there is a power button, okay?

It looks like an x ​​with a little filled square in the corner, okay? So if you type that, for example, if you go, let me turn this on... if you go to Seven, then you press the power button to the right and that little box will appear. So this is what I wrote, okay. 7, then I'll write that x with a box now if we write something like 3, then three now appears in the right power, so seven is the base and three is the pound. You press equals and you get 343. Okay, so seven cubed is 343, this is what we're going to do.

I want us all to get a bunch of different numbers, okay? So, raise your hand who has a proper calculator and not another phone, okay? So here's what I'm going to do: I'm going to write a bunch of things down correctly and then I want us to figure out what their value is. Well? so value y and Hmm. Surely this way I will make all this clear. I'll think of a bad name in a second, okay, so it's really important. We do the, um, the calculations with your calculator. So raise your hand again so I can assign some numbers, okay?

Everybody raise your hands so I can see, okay... okay. jagr, you get the hardest one. Well, you get one from the power of one, okay? Keep your hand up, Kyle, you'll get 0.9 to the power of 0.9. (kyle: that's fine with me, yeah) Okay, Jack, you can do 0.8 to the power of 0.8. Do you see that I'm doing well? Tom - .7. It was good, okay? Alright, let's start from the top and remember the three decimals again. Well, Jagr, make sure you read it carefully so you don't miss any details. What did you get? (jagr: 1) one but some decimal? Okay, so I'm going to go to zero zero zero, okay?

There you go. Alright! Kyle you're next! .9! (Kyle: 0.909) 0.909?! Okay, well, are you going to shoot next zero point eight three six Eight three six, okay, Tom Seven seven nine, okay? I want to pause for a second before we get to Lucas's number. Just look up for a second and watch what's happening, just watch what's happening, okay? I'm trying to establish a pattern. Can you see the numbers again? But trying to get this point right, try closer and closer at this point if you had to guess at this point. What would you think would happen when we reach zero power zero?

How does it look like it's going to get there? It seems like they are going to zero, right? They were falling.. hmm.. well let's continue! who's turn is it? Luke? lucas: 0.736 0.736 okay, who's next? - Nelson - Yes? Nelson: zero point sev.. amm Nelson: seven zero seven.. teacher: seven zero seven? Okay, okay, now to Bill *Bill starts mumbling Okay, okay, now we'll pause for a minute to move on to the next number. The people who were suspicious before now should have caused even more suspicion because he watches. The numbers are going down, right? But not by much.

Look at these guys, right? They're getting pretty close. They are decreasing, but they are slowing down. Yes it's correct. Well, actually we call that is exactly what it is. We call this exponential growth, but what we're seeing appears to be exponential decay, but anyway, that's a task. It gets even more interesting, okay? Who had point three, yeah? 696 696...what?! What happened? Why am I slowing down or even slowing down? It's gone the other way! Turn around. Well? Let's keep going! point two (0.2) Wait... two four... we go up again. We are going up. Well. Chloe forward Chloe: 0.793 okay, now this is interesting, I'm getting close.

I know I'm getting there, but I'm number one. I'm not there number two, you see how we were slowing down from here to here, right? These gaps are getting very, very small, but now that I turned around, the gaps are getting bigger. Look, that's that gap like point three, this gap like point seven, right? So what is going on? Well? Now? We need more numbers, so we need your calculators again here the numbers. I'm going to suggest Let's go. Let's get half of this. What is half of point one? I think it's .05, isn't it? Jarrod, they fooled you by making one to the power of one.

So why don't you do the 0.05 to the power of 0.05? Well? Hey Kyle. You want to do? Let's make it smaller again. let's go from .02 to 0.2. Okay, Jack, join 0.01. Well, I'll tell you what's next in a second. Jack can tell me which one is yours. Jarrod: 0.860 zero point eight six zero Hmm, still cooking, increasing yes Kyle: 0.924 0.924, still increasing, yes Jack: 0.954 zero Point nine five four Hmm Okay, this time we're going towards zero, but in smaller portions, okay? ? I want to get there faster. Well, this time Tom. What I want you to put in is so other people can see if they get the answer faster feeling like we're going ten times Smaller at zero point zero zero one, okay?

What do you get? Yes 993? Okay, let's go again. Let's go back to being ten times smaller, 0.0001 to the power of 0.0001. What does that equate to? Did you understand Lucas? Yes Yes. Ah, now just because we got to nine. What are, out of curiosity, what are the other decimals? Okay, okay, Ali Ali, can you do the next one we do? 0.00001 is four zeros, and to the power of zero point zero zero zero and one, yes. four zeros one yes four zeros and then one what do we get? And this time I'm going to need more decimals. What happened with the breakup? ah You missed a 1 there, oh Go back to the beginning Ok Math error what zero point did you get?

Yeah? I'm (reading the number) good, great. I'll go there. Yeah, okay, five zero is five... Oops, sorry, one two three four five, okay? Okay, okay sign on the spot, okay, now. This is interesting. If I'm not mistaken, if you can get far enough with your calculator. I think you will convince your calculator. Then it comes to one. Suspect. Well. I can't, well, you can sit down, you can try and prove it. But you can see we've done two things, okay? First, we saw him fall, right? But then he turned around. It started to increase like this.

But then it didn't increase forever. , really Okay, did anyone get a one? How many do I have to put? I use it. You can spam yourself. I continued with your rows until it's right. Five minutes ago. 20 20 zeros can't go, so by the way, what have you done? To get one, all you've done is run out of nines on your calculator. That's all yours, like we have a longer calculator or maybe some of your phones have more digits in them that don't. It doesn't have notation for that one and also The 9 don't repeat forever, there are other numbers, so you can't use a repeater, so therefore we never get to 0, not 1 0, but the well.

It's not entirely true either, but the best definition you get is where it seems to go towards 1, okay? That's the best we can say, but I don't bother. Yeah, well, we'll see that it starts to fail if you go further because there are negative aspects. It doesn't work at all. Well, look what happens. I like this answer. This is an answer We decided on the numbers and you know, they start with our definition, so that's what makes the most sense in this context. Well, let me finish by returning to this idea. What everyone is doing with their calculators is called taking a limit?

There is the idea that something is approaching a certain number. We can never get there, but you can get very close. So if you're the curious type, you can write this and surprise your friends with it, okay? It's a bit of a strange notation, but I'll explain what it means in a second. You should type it in and then it will look like this, okay, this, what does it mean? Or it means seeing this right, it means that when x gets closer to 0, you can see the error x goes there, right as x gets closer and closer. and closer to 0 What's up... with... this guy here? this.. that to the power of itself And the answer is, go to that!

That's what it does, okay? He never gets there, but that's where he'll head if he could get there. That's what happened, okay? So I hope that gives you a conclusive answer. So if you search the internet, which I know some of you did. So you will find that this is the conclusion. Yes. Do you mean why it rotates here? That's a good question, and I don't have an answer for that, at least I mean I don't have one right now, probably because I haven't thought about it much to be honest. I wonder, I mean there are always always reasons for everything.

So maybe we can find out where they're going.