Evaluating fractional exponents: fractional base

fractional exponents going to be a little daunting at first so it never hurts to do as many examples as possible so let's do a few what if we had 25 over 9 and we wanted to raise it to the 1/2 power so we're essentially just saying well what is the principal square root of 25 over 9 so what number times itself is going to be 25 over 9 well we know 5 times 5 is 25 and 3 times 3 is 9 so why don't we just go with 5 over 3 because notice if you have 5 over 3 times 5 over 3 that is going to be 25 over 9 or another way of saying this that 5 over 3 squared is equal to 25 over 9 so 25 over 9 to the one-half is going to be equal to 5/3 now let's escalate things a little bit let's take let's take a really hairy one let's raise let's raise 81 over over to 56 81 over 256 to the negative 1/4 power I encourage you to pause this and try this on your own so what's going on here this negative the first thing I always like to do is I want to get rid of this negative in the exponent so let me just take the reciprocal of this and raise it to the positive so 1 so I could just say that this is equal to 256 over 81 to the 1/4 power and so now I can say well what number times itself times itself times itself is going to be equal to 256 and what number times itself times itself times itself did I say that four times well what number if I take four of them and multiply do I get 81 and one way to think about it this is going to be the same thing and we'll talk about this in more depth later on when we talk about exponent properties but this is going to be the exact same thing as 256 to the one fourth over 81 to the 1/4 you in fact saw it over here this over here was the same thing as the square root of 25 over the school 9 or 25 to the 1/2 over 9 to the 1/2 so we're just doing that over here so one we still have to think about what number this is so and this is a little bit of there's no easy way to do this you kinda have to just play around a little bit to come up with it but 4 might jump out at you if you recognize that 16 times 16 is 256 we know that 4 to the 4th power or you you're about to know this is 4 times 4 times 4 times 4 and 4 times 4 is 16 times 4 is 64 times 4 is equal to 256 so 4 to the fourth is 256 or we could say 4 is equal to 256 to the 1/4 power fair enough now what about 81 well 3 might jump out at you we know that 3 to the 4th power is equal to 3 times 3 times 3 times 3 which is equal to 81 so 3 is equal to 81 to the 1/4 so this top number 256 to the 1/4 that's just 4 81 to the 1/4 that is just 3 so this right over here is going to be equal to 4/3