1 and -1 to different powers

let's think about exponents with ones and zeroes so let's take the number one and let's raise it to the 8th power so we've already seen that there's two ways of thinking about this you could literally view this as taking eight ones and then multiplying them together so let's do that so you have one two three four five six seven eight ones and then you're going to multiply them together and if you were to do that you would get well one times one is one times one it doesn't matter how many times you multiply one by one you are going to just get one you are just going to get one and you could imagine I did it eight times I multiplied eight ones but even if this was 80 or if this is 800 or if this was eight million if I just multiplied one if I had eight million ones and I multiplied them all together it would still be equal to one so 1 to any power is just going to be equal to one and would say hey what about 1 to the 0th power 1 to the 0th power well we've already said anything to the zero power except for zero that's we're going to it's actually up for debate but anything to the zero power is going to be equal to one and just as a little bit of intuition here you could literally view this as our other definition of exponent X potentiation which is you start with the 1 and this number says how many times you're going to multiply that 1 times this number so 1 times 1 0 times is just going to be 1 and that was a little bit clearer when we did it like this where we said 2 to the let's say fourth power 2 to the fourth power is equal to this was the other definition of exponentiation we had which is you start with a 1 and then you multiply it by 2 4 times so times 2 times 2 times 2 times 2 which is equal to let's see this is equal to 16 so here if you start with a 1 and then you multiply it by 1 0 times you're still going to have that one right over there that's why anything that's not 0 to the 1 power is going to be equal to 1 now let's try some other interesting scenarios let's start try some negative numbers so let's take negative 1 and let's first raise it let's first raise it to the zero power so once again this is just going based on this definition this is starting with a 1 and then multiplying it by this number zero times well that means we're just not going to multiply it by this number so you're just going to get a 1 let's try negative 1 let's try negative 1 to the first power well anything to the first power you could view this and I like going with this definition as opposed to this one right over here if we were to make them consistent if you were to make this definition consistent with this you would say hey let's start with a 1 and then multiply it by 1 8 times and you're still going to get a 1 right over here but let's do this with a negative 1 so we're going to start with a 1 and then we're going to multiply it by negative 1 one time times negative 1 and this is of course going to be equal to negative 1 now let's take negative 1 and let's take it to the second power we often say that we are squaring it when we take something to the second power so negative 1 to the second power well we could start with a 1 we could start with a 1 and then multiply it by negative 1/2 times multiply it by negative 1 multiply it by negative 1 twice and what's this going to be equal to and once again by our old definition you could also just say hey ignoring this one because that's not going to change the value we took two negative ones and we're multiplying them well negative 1 times negative 1 is 1 and I think you see a pattern forming let's take negative 1 let's take negative 1 to the third power to the 3rd power what's this going to be equal to well by this definition you start with a 1 and then you multiply it by negative 1 3 times so negative 1 times negative 1 times negative 1 or you could just think of it as you're taking 3 negative ones and you're multiplying it because this one doesn't change the value and this is going to be equal to negative 1 times negative 1 is positive 1 times negative 1 is negative 1 so you see the pattern 1 negative 1 to the 0 power is 1 negative 1 of the first power is negative 1 then you multiply by negative 1 and getting it positive one then you multiply by negative one again to get negative one and the pattern you might be seeing is is if you take negative one to an odd power to an odd power you're going to get negative one and if you take it to an even power you're going to get one you're going to get one because of negative times a negative is going to be the positive and you're gonna have an even number of negatives so that you're always going to have negative times negatives so this right over here this is even even is going to be positive one and you can see that if you went to negative one to the fourth power negative one to the fourth power well you could start with a 1 and then multiply it by negative one four times so negative 1 times negative one times negative one times negative one which is just going to be equal to positive one so if someone were to ask you if someone were to ask you we already established that if you someone were to take one to the I don't know one millionth power to the one millionth power this is just going to be equal to one it's just going to be equal to one if someone told you let's take negative one and raise it to the 1 millionth power one millionth power well 1 million is an even number so this is still going to be equal to positive one but if you took negative one if you took negative 1 to the 999,999 power this is an odd number so this is going to be equal to this is going to be equal to negative one