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# Exponentiation warmup

## Video transcript

hi Sal hey Britt how are you good looks like we have a a game going on here not a game yeah I kind of kind of a challenge question for you what I did is I put one grain of rice in the first square that's right there's 64 squares on the board yep and at each consecutive square I doubled the amount of rice mm-hmm how much rice do you think would be on this square on that square so let me let me think about it a little bit actually I'm going to fix it so here you have one and we multiply that times two so this is going to be 2 times 2 I don't know 2 times 1 what am i doing now this is 2 times that 1 so this is 2 times 2 now this is 2 times that so this is okay we're starting to take a lot of tools here multiplying them together so this is 2 times 2 up front right side ways times 2 so this one is going to be 5 twos multiplied together this is going to be 6 twos multiplied together this is going to be 7 2's multiplied together 8 2's multiplied together so nine twos 10 11 12 13 so to all of this stuff multiplied together 8,192 grains of rice is what we should see right over here and you know I had fun last night I was up late but there you go did you really count out 8,192 grains of rice more or less okay let's just say you did what if we just went you know four steps ahead and how much rice would be here four steps ahead so we're going to multiply by two then multiply by two again then multiply by two again then multiply by two again so it's this number times the C 2 times 2 is 4 times 2 is 8 times 2 is 16 so it's going to get us like 120 like hundred thirty thousand or around there's 130 1672 you got a lot of time last night we're not even halfway across the board yet we're not a I mean this is this is this is a lot of that's a lot of rice there that you could throw a party what about the last square this is 63 steps we're going to take we're going to take two times two and we're going to do 63 of those so this is going to be a huge number actually it would be neat if there was a notation for that I didn't 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 but I want a question I mean this was a little bit of a pain for me to write all of these two so was this if I were the mathematical community mm-hmm I would want some type of notations you kind of got on it here I like this dot dot dot and the 63 this you know I understand this yeah you can understand this but this is still a little bit it's a little bit too much what if instead we just wrote mathematicians love being efficient right they were they're lazy they have things to do they have to go home and count grains of rice right this is No yeah so that is tack 63 twos and multiply them all together this is the first square on our board we have one one grain of rice yeah and when we double it we have two grains of rice yep and we double it again we have four and I'm thinking this this is similar to what we were doing it's just represented differently yeah well I mean this one the one you were making right every time you you're kind of adding these more these popsicle sticks you're kind of branching out you know one popsicle stick now becomes two popsicle sticks and then you keep doing that one popsicle stick becomes two but I get two of them so here you have one now you have one times two now each of these two branch into two so now you have two times two or you have four popsicle sticks every stage every branch you're multiplying by two again I basically just continue splitting just like a tree does yep now I can really see what 2 ^ 3 looks like and that's what we have here 1 times 2 times 2 times 2 which is 8 this is 2 to the 3rd power when I see 2 to the power of something let's just say n mmm n could also be number of steps up this tree I could think about it that way yeah you could view it I guess one way to think about is how many times you've branched but that one that tree there is actually even new morning I don't think this counts because again like this branch is 4 times at each branch well I guess why not I mean well it's different it's not going to be 2 anymore so this 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 branch like this and now each of those so now you've branched twice so now this is 4 to the second power so yeah this the base or what's called the base when you're making an exponent this 4 right over here this is how many times how many new branches each of the branches turn into at each of these I guess in junctions you can say it's calling Junction junctions you haven't you haven't branched yet here you branched once and here you've branched twice but this is this is interesting this is also why when I look at a tree you know there's thousands of leaves but just one trunk and when you actually go up and you look at each you look inside the tree it only branches you know three or four times and that shows the power of exponential growth yes