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# Negative exponents

CCSS.Math:

## Video transcript

we already know that two to the fourth power can be viewed as starting with a 1 and then multiplying it by 2 4 times so let me do that so times 2 times 2 times 2 times 2 and that will give us let's see 2 times 2 is 4 8 16 so that will give us 16 now I will ask you a more interesting question what do you think 2 to the negative negative 4 powers and I encourage you to pause the video and think about that well you might be tempted to say oh well maybe it's negative 16 or something like that but remember what the exponent operation is trying to do this is one way of viewing it is this is telling us how many times are we going to multiply 2 times negative 1 but here we're gonna we're gonna multiply negative 4 times what what is negative traditionally me negative traditionally means the opposite so here this is how many times you're going to multiply maybe when we make it negative this says how many times we're gonna start it with a 1 how many times are we going to divide by 2 so let's think about that a little bit so this could be viewed as 1 times and we're going to divide by 2 4 times Well dividing by 2 is the same thing as multiplying by 1/2 so we could say that this is 1 times 1/2 times 2 me student 1 colors can take so 1 times 1/2 times 1/2 times 1/2 times 1/2 notice multiplying by 1/2 4 times is the exact same thing as dividing by 2 4 times and in this situation this would get you a 1/2 well 1 times 1/2 is just 1/2 times 1/2 is 1/4 times 1/2 is 1 8 times 1/2 is 1 over 16 and so you probably see the relationship here if you're this is essentially you're starting with the one you're dividing by 2 4 times you could also say you could also say that 2 - I'm gonna do the same colors 2 to the negative 4 2 to the negative 4 is the same thing as 1 Oh for two to the fourth power one over two to the fourth power let me color code it nicely so you realize what the negative is doing so this is negative right over here let me do that in a better color other than magenta something that jumps out so this negative right over here this is what's causing us to go one over so two to the negative four is the same thing based on the way we've defined it just up right here as one over the order is the reciprocal of 2 to the fourth or 1 over 2 to the fourth and so you could view this as being 1 over 2 times so 2 times 2 times 2 times 2 if you just do 2 to the fourth as taking 4 2s and multiplying them or if you use this idea right over here you could view it as starting with a 1 and multiplying it by 2 4 times either way you are going to get 1 over 1 over 16 1 over 16 so let's do a few more examples of this just so that we make sure things are clear to us so let's try let's try 3 to the negative third power so remember whenever you see that negative what my brain always does is say I need to take the reciprocal here so this is going to be equal to I'm gonna highlight the negative again this is going to be 1 over 3 to the third power 1 over 3 to the third power which would be equal to well 1 over 3 times 3 or you could say 1 over 3 times 3 times 3 or 1 times 3 times 3 times 3 is going to be 27 so this is going to be 127th let's try another example I'll do 2 or 3 more so let's take let's take a negative number to a negative exponent just to see if we can confuse ourselves so let's take the number negative 4 negative 4 and let's take it I don't want my numbers to get too big too fast so let's take so let's just take negative 2 let's take negative 2 and let's take it to the negative 3 power negative I want to make my negatives in magenta negative 3 power negative negative three power so at first this might be daunting do the negatives cancel and that will just be the remnants is in your brain that they're trying to think of multiplying negatives do not apply that here remember you see a negative exponent that just means the reciprocal of the positive exponent so 1 over negative 2 negative 2 to the third power so the positive third power and this is equal to this is equal to 1 over negative 2 negative 2 times negative 2 times negative 2 times negative 2 or you could view it as 1 times negative 2 times negative 2 times negative 2 which is going to give you 1 over negative 8 or negative negative 1 eighths let me scroll over a little bit I don't want to have to start squinching things so this is equal to negative 1/8 let's do one more example just in an attempt to confuse ourselves let's take let's take I don't know let's take 5/8 and raise this to the negative negative 2 power negative 2 power so once again this negative way I got a fraction is a negative here remember this just means 1 over 5/8 to the 2nd power so this is just going to be the same thing as 1 over five eighths 1 over 5/8 squared 1 over 5/8 squared which is going to be the same thing which is going to be the same thing so this is going to be equal to trying to color-code it 1 over 5/8 times 5/8 which is 25 over 60 25 over 64 1 over 25 over 64 is just going to be 64 over 25 so one another way to think about it is you're going to take the reciprocal of this and raise it and raise it to the positive exponent so another way you could have thought about this another way you could have thought about this is 5/8 to the negative 2 power I'm just take the reciprocal of this eight fish and raise it to the positive to power so all of these statements are equivalent and that would have applied even when you are dealing with non fractions as your base right over here so too you could say well this is going to be the same thing two to the negative four is going to be the same thing as taking my reciprocal so this is going to be the same thing as taking the reciprocal of 2 which is which is 1 over 2 and raising it to the positive to the positive 4 power