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CCSS.Math:

if we think about something like 2/3 power we could view this as taking three twos and multiplying them together so two times two times two or equivalently we could say well this is the same thing as taking a 1 and then multiplying it by 2 3 times so let's see let's just go with this definition right over here and this of course is going to be equal to 8 now what would based on this definition I just did what would 2 to the second power be well this would be 1 times 2 twice so 1 times 2 times 2 which of course would be equal to 4 what would 2 to the first power be well that would be 1 and we would multiply it by 1/2 1 times 2 which of course is equal to 2 now let's ask ourself an interesting question based on this definition of what an exponent is what would 2 to the 0th power be and encourage you to just think about that a little bit if you were the mathematics community how would you define 2 to the 0 power so it's consistent with everything that we just saw well the way we just talked about it we just said exponentiation is you start with the 1 and you multiply it by the base zero times so we're not going to multiply it by any two so we're just going to be left with a 1 so does this make sense that 2 to the 0 power is equal to 1 let's think about it another way let's think about another way let's do it let's do a different base now that was with 2 but let's say we have I don't know let's say we have 3 and I could say you know 3 to the 4th power that's 3 times 3 times 3 times 3 which is going to be equal to 81 actually let me just write that down let me just write down that this is going to be equal to 81 if I said 3 to the third power that's 3 times 3 times 3 which is 27 3 to the second power is equal to 9 3 to the first power is equal to 3 do you notice a pattern every time we decrease the exponent here by 1 so we want 3 to the fourth and now we go through the third what happened what happened to the value well going from 81 to 27 we divided by three and that makes sense because we're multiplying by one less three so we divide by three to go from 81 to 27 we divide by three again if our exponent goes down by one and we divide by three again when we go from nine to three divided by three so based on this what do you think three to the zero power should be well the pattern is every time we decrease our exponent by one we divide by the base and so we should divide by three again would be the logic if we follow that pattern and so three divided by three you would get us one again so I know it might seem a little bit counterintuitive that something to the zeroth power is going to be equal to one but this is how the mathematics community has defined it because it actually makes a lot of sense either if you view an exponent as taking a 1 and multiplying it by the base the exponent number of times so I'm going to multiply one by two three times or if you just follow this pattern that look start you know every time you decrease the exponent by one you're going to be dividing by the base either of those would get you to the conclusion that 2 to the 0 power is 1 or 3 to the 0 power is 1 or frankly any number to the 0 power is 1 so any number so if I have any number let's say I have some number a to the 0 power this is going to be equal to 1 I have an interesting question for you and let's just say this is this is the case when a does not equal 0 I'll leave you a little bit of a puzzle for you to think about what do you think 0 to the zeroth power what should to the 0 power be and what's interesting what's interesting about 0 to the 0 power is you'll get a different answer if you use this technique versus if you use this technique right over here this technique would actually get you to being 1 while this technique would have you divided by 0 which we don't know how to do anyway I'll leave you there to ponder the mysteries zero to the zeroth power