The 0 & 1st power

what I want to do in this video is think about exponents in a slightly different way that will be useful for different contexts and also go through a lot more examples so the last video we saw that taking something to an exponent means multiplying that number that many times so if I had the number negative 2 and I want to raise it to the third power this literally means taking 3 negative twos so this literally means taking 3 negative 2 so negative 2 negative 2 and negative 2 and then multiplying them and then multiplying them so what's this going to be well let's see negative 2 times negative 2 is positive 4 and then positive 4 times negative 2 is negative 8 so this would be equal to this would be equal to negative 8 now another way of thinking about exponents instead of saying you're just taking 3 negative 2s and multiplying and this is a completely reasonable way of viewing it you could also view it as this isn't this is a number of times you're going to multiply this number times one so you could completely view this as being equal to so you're going to start with a 1 and you're going to multiply 1 times negative 2 3 times so this is times negative 2 times negative 2 times times negative 2 so clearly these are the same number here we just took this and we're just multiplying it by 1 so you're still you're still going to get negative 8 and this might be a slightly more useful idea to get an intuition for exponents especially when you start taking things to the 1 or 0 power so let's think about that a little bit what is 2 what is positive 2 to the based on this definition to the 0 power going to be equal 2 to the 0 power going to be equal to well we just said this says how many times are you going to multiply 1 times this number so this literally says I'm going to take a 1 I'm going to take a 1 and I'm going to multiply it by 2 zero times well if I want to multiply it by two zero times that means I'm left with the 1 so 2 to the 0 power is going to be equal to 1 and actually any nonzero number to the 0 power is 1 by that same rationale and I'll make another video that will also give a little bit more intuition intuition on there that might seem very counterintuitive but it's based on one way of thinking about it is thinking of an exponent as this and this will also make sense if we start thinking of what 2 to the 1st power is so 2 2 to the 1st power so let's go to this this definition we just gave her the exponent we always start with a 1 and we multiply it by the 2 one time so 2 is going to be 1 we're only going to multiply it by the two I'll use this for multiplication I use the dot we're going to multiply it by 2 one time so 1 times 2 well that's clearly just going to be equal to 2 and any number to the first power is just going to be equal to that number and then we can go from there and you'll of course see the pattern if we say what 2 squared is 2 squared well based on this definition we start with a 1 and we multiply it by 2 2 times so times 2 times 2 is going to be equal to 4 and we've seen this before you go to 2 to the third 2 to the third power you start with a 1 start with a 1 and then multiply it by 2 3 times so times 2 times 2 times 2 this is going to give us positive positive 8 and you probably see a pattern here every time we multiply by 2 or every time I should say we raise 2 to a 1 more power we are multiplying by 2 notices you to go from 2 to the 0 2 to the 1 we multiplied we multiplied by 2 I'll use a little X for the multiplication symbol now they'll across and then to go from 2 to the first power to 2 to the second power we multiply by 2 multiplied by 2 again and that makes complete sense because this is literally telling us how many times are we going to take this number and how many times I'm going to take one and multiply it by this number and so when you go from two to the second part of to the third you're multiplying by two multiplying by two one more time and this is another intuition of why something to the zero power is equal to one if you were to go backwards if say we didn't know what two to the zero power is and we were just trying to figure out what would make sense when we go from two to the third power to two to the second we'd be dividing by two we're going from eight to four then we divide by two again to go from ii ii ii ii ii to the first and then maybe it seems like we should just divide by two again for two from going from 2 to the first to two to the zero and that would give us 1