Exponentiation warmup

Voiceover:Hi Sal. Voiceover:Hey Britt. Voiceover:How are you? Voiceover:Good, looks like we have a game going on here. Voiceover:Not a game. Yeah, kind of a challenge question for you. What I did is, I put 1 grain of rice in the first square. Voiceover:That's right. Voiceover:There's 64 squares on the board. Voiceover:Yup. Voiceover:And in each consecutive square I doubled the amount of rice. Voiceover:Mm hm. Voiceover:How much rice do you think would be on this square? Voiceover:On that square? Let me think about it a little bit. Actually, I'm going to take some ... Here you have 1 and we multiply that times 2, so this is going to be 2 times 2. No, no 2 times 1, what am I doing? Now this is 2 times that one so this is 2 times 2. Now this is 2 times that. So this is ... Okay, we're starting to take a lot of 2's here and multiplying them together. So this is 2 times 2 ... I'm trying to write sideways. Times 2. This one is going to be 5, 2's multiplied together. This is going to be 6, 2's multiplied together. This is going to be 7, 2's multiplied together. 8, 2's multiplied together. 9, 2's. 10, 11, 12, 13. So all of this stuff multiplied together. 8,192 grains of rice is what we should see right over here. Voiceover:And you know, I had fun last night and I was up late, but there you go. Voiceover:Did you really count out 8,192 grains of rice? Voiceover:More or less. Voiceover:Okay. Let's just say you did. Voiceover:What if we just went, you know, 4 steps ahead. How much rice would be here? Voiceover:4 steps ahead, so we're going to multiple by 2, then multiple by 2 again, then multiply by 2 again, the multiply by 2 again. So it's this number times ... Let's see, 2 times 2 is 4. Times 2 is 8, times 2 is 16. So it's going to get us like 120, like 130,000 or around there. Voiceover:131,672. Voiceover:You had a lot of time last night. We're not even halfway across the board yet. Voiceover:We're not. Voiceover:This is a lot of ... That's a lot of rice, there. You could throw a party. Voiceover:What about the last square? This is 63 steps. Voiceover:We're going to take 2 times 2 and we're going to do 63 of those. So this is going to be a huge number. And actually, it would be neat if there was a notation for that. Voiceover:I didn't count this one out but it is the size of Mount Everest, the pile of rice. And it would feed 485 trillion people. Voiceover:But I have one question. I mean, you know, this was a little bit of a pain for me to write all of these 2's. Voiceover:So was this. Voiceover:If I were the mathematical community I would want some type of notation. Voiceover:You kind of got on it here. I like this dot, dot, dot and the 63. This I understand this. Voiceover:Yeah, you could understand this but this is still a little bit ... This is a little bit too much. What if, instead, we just wrote ... Voiceover:Mathematicians love being efficient, right? They're lazy. Voiceover:Yeah, they have things to do. They have to go home and count grains of rice. Voiceover:Right. (laughter) Voiceover:Yeah. So that is, take 63, 2's and multiply them all together. Voiceover:This is the first square on our board. We have 1 grain of rice. And when we double it we have 2 grains of rice. Voiceover:Yup. Voiceover:And we double it again we have 4. I'm thinking this is similar to what we were doing, it's just represented differently. Voiceover:Yeah, well, I mean, this one, the one you were making, right, every time you were kind of adding these popsicle sticks, you're kind of branching out. 1 popsicle stick now becomes 2 popsicles sticks. Then you keep doing that. 1 popsicle stick becomes 2 but now you have 2 of them. So here you have 1, now you have 1 times 2. Now each of these 2 branch into 2, so now you have 2 times 2, or you have 4 popsicle sticks. Every stage, every branch, you're multiplying by 2 again. Voiceover:I basically just continue splitting just like a tree does. Voiceover:Yup. Voiceover:Now I can really see what 2 to the power of 3 looks like. Voiceover:And that's what we have here. 1 times 2 times 2 times 2, which is 8. This is 2 to the third power. Voiceover:When I see 2 to the power of something, let's just say N. N could also be number of steps up this tree. I could think about it that way. Voiceover:Yeah, you could view it ... I guess one way to think about it is how many times you've branched. But that one, that tree there, is actually even more interesting. Voiceover:I don't think this counts because, again, this branches 4 times at each branch. Voiceover:Well I guess why not? It's different. It's not going to be 2 anymore. So the first one where you haven't branched yet, this is going to be 4 to the 0 power. You've had no branches yet. This, you branched once so now this is 4 to the first power. You have 4 branches now. Voiceover:Oh, I like this. Voiceover:And now each of those. So now you've branched twice. So now this is 4 to the second power. So yeah, the base, or what's called the base when you take an exponent, this 4 right over here. This is how many new branches each of the branches turn into at each of these, I guess, junctions you could say. Voiceover:Let's call them junctions. Voiceover:Junctions. You haven't branched yet. Here you've branched once, and here you've branched twice. Voiceover:This is, this is interesting. This is also why when I look at a tree there's thousands of leaves but just 1 trunk. And when you actually go up and you look inside the tree it only branches 3 or 4 times. Voiceover:And that shows the power of exponential growth. Voiceover:Yes. (laughs)