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Evaluating quotient of fractional exponents

CCSS.Math: ,

Video transcript

let's see if we can figure out what 256 to the 4/7 power divided by 2 to the 4/7 power is and like always pause the video and see if you can figure this out alright let's work through this together and at first you might find this kind of daunting especially when you see something like 2 to the 4/7 power is that even that's not going to be a whole number how do i how do I do this especially without a calculator and I should have said do this without a calculator but then the key is to to see to see that we can use our exponent properties to simplify this a little bit so that we can do this on paper and the main property that might jump out at you is if I have something if I have if I have X to the a power over Y to the a power this is the same thing as x over Y to the a power and our situation right over here 256 would be X 2 would be Y and then a is 4 7 so we can rewrite this this is going to be equal to this is equal to 256 over 2 to the 4/7 power and so this is nice we're already able to simplify this because we know 256 divided by 2 is 128 so this is 128 to the 4/7 power now this might also seem a little bit difficult how do I raise 128 to a fractional power but we just have to remind ourselves this is the same thing this is the same thing as 128 to the 1/7 power then raised to the 4th power we could also view it the other way around we could say that this is also 128 to the 4th to the 4th power and then raise that to the 1/7 but multiplying 128 four times that's going to be very computationally intensive and then you have to find the seventh root of that that seems pretty difficult so we don't want to go in that way but if we can get the smaller number first what is 128 to the seventh to the one seventh power then that might be easier to raise to the fourth power now when you look at this and knowing that probably the generator in this case I'm the person who presented with you is telling you that you're not going to use a calculator it's a pretty good clue that all right this is probably going to be a this is probably going to be something that I can figure out on my own and you might recognize 128 as a power of two and maybe two to the seventh is 128 and we can verify that so let's see two to the first is 2 4 8 16 32 64 128 2 times 2 is 4 times 2 is 8 times 2 is 16 times 2 is 32 times 2 is 64 times 2 is 128 so 2 to the seventh power is equal to 128 or another way of saying this exact same thing is that 128 128 is equal to or 128 to the one seventh power is equal to 2 or you could even say that the seventh root the seventh root of 128 is equal is equal to two so we can simplify this this is two so our whole expression is now just two to the fourth power well that's just 2 times 2 times 2 times 2 so that's 2 to the fourth power 2 to the fourth power which is just going to be equal to 16 that's 2 times 2 times 2 times 2 right over there and so we're done this crazy complicated looking expression it is simplified to 16