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Radicals and rational exponents — Basic example

Video transcript

which of the following values is equal to the value above and we got this this expression here with a bunch of negative fractional exponents and at first you might say oh how do I deal with this maybe you know I don't know what the fifth the fifth root of three is much less than negative fifth root and this 1/3 to the negative 2/5 how do I simplify this and the key realization here is that 1/3 is the same thing as 3 to the negative 1 power one-third is the same thing as 3 to the negative 1 power and if we do that I suspect that we're going to get be able to get similar bases here let me rewrite the whole thing so it's gonna be 3 to the negative 1/5 times instead of writing 1/3 I could write 3 to the negative 1 power to the negative 2/5 power and now we could just use up our straight exponent rules to simplify things a little bit so this business right over here that I'm squaring in this orange color if I raise something to an exponent and then raise it again to another exponent that's going to be the same thing as taking our original race and raising it to the negative 1 times the negative 1 times negative 2/5 power so I can raise it if I raise it to the negative 1 and then to the negative 2/5 that's the same thing as raising 3 to the negative 1 times negative 2/5 power and so and this over here is still going to be 3 to the negative 1/5 and so this is going to be equal to 3 to the negative 3 to the negative 1/5 times now negative 1 times negative 2/5 that's going to be positive 2 6 so times 3 to the 2/5 and now we have a situation where we have the same base we have the product of 3 to the negative 1/5 times 3 to the positive 2/5 this is going to be equal to we can take our base 3 it's going to be 3 to the negative 1/5 plus 2/5 power plus 2/5 power if you have the same base the product of that base raised to one exponent on that same base raised to another exponent that's the same thing as that base raised to the some of those exponents of classic exponent property encourage you to go Kahn Academy if this was looking for it or if you need some review but now this is pretty straightforward this is going to be equal to this is going to be equal to 3 to the negative 1/5 plus 2/5 is just 1/5 and there you have it 3 to the 1/5 power