Powers of whole numbers
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- If we think about something like two to the third power, we could view this as taking three twos and multiplying them together, so two times two times two, or equivalently, we could say this is the same thing as taking a one and then multiplying it by two three times, so actually, let's just go with this definition right over here, and this, of course, is going to be equal to eight. Now what would, based on this definition I just did, what would two to the second power be? Well, this would be one times two twice, so one times two times two, which, of course, would be equal to four. What would two to the first power be? Well, that would be one, and we would multiply it by one two, one times two, which, of course, is equal to two. Now let's ask ourself an interesting question. Based on this definition of what an exponent is, what would two to the zeroth power be? I encourage you to just think about that a little bit. If you were the mathematics community, how would you define two to the zero power so it is consistent with everything that we just saw. Well, the way we just talked about it, we just said exponentiation is you start with a one and you multiply it by the base zero times, so we're not gonna multiply it by any two, so we're just gonna be left with a one. So does this make sense that two to the zero power is equal to one? Well, let's think about it another way, and let's do a different base. That was with two, but let's say we have three and I could say three to the fourth power, that's three times three times three times three which is going to be equal to 81, and let me just write down that this is going to be equal to 81. If I said three to the third power, that's three times three times three which is 27. Three to the second power is equal to nine. Three to the first power is equal to three. Do you notice a pattern every time we decrease the exponent here by one? We want three to the fourth, and now we go three to the third. 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, divide 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 would get us one again. So I know it might seem a little bit counter intuitive 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 one and multiplying it by the base the exponent number of times, so I'm gonna multiply one by two three times, or if you just follow this pattern, 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 two to the zero power is one, or three to the zero power is one, or frankly, any number to the zero power is one. So if I have any number, let's say I have some number a to the zero power, this is going to be equal to one. Now I have an interesting question for you, and let's just say this is the case when a does not equal zero. I'll leave you a little bit of a puzzle for you to think about. What do you think is zero to the zeroth power? What should zero to the zero power be? And what's interesting about zero to the zero 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 one, while this technique would have you divide by zero, which we don't know how to do. Anyway, I'll leave you there to ponder the mysteries of zero to the zeroth power.