Radicals and rational exponents — Basic example

- [Instructor] Which of the following values is equal to the value above? And we've got this expression here with a bunch of negative fractional exponents. And at first, you might say okay, how do I deal with this? I don't know that the fifth root of three is, much less the negative fifth root, and this one third to the negative two fifths. How do I simplify this? And the key realization here, is that one third is the same thing as three to the negative one power. One third is the same thing as three to the negative one power. And if we do that, I suspect that we're gonna be able to get similar bases here. Let me rewrite the whole thing. So, it's gonna be three to the negative one fifth times, instead of writing one third, I can write three to the negative one power, to the negative two fifths power. And now we can 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 raise and raising it to the negative one times the negative one times negative two fifths power. So if I raise it to the negative one, and then the negative two fifths, that's the same thing as raising three to the negative one times negative two fifths power. And so this over here is sill going to be three to the negative one fifth. And so this is going to be equal to three to the negative one fifth times, now negative one times negative two fifths, that's going to be positive two fifths. So we times three to the two fifths. And now we have a situation where we have the same base. We have the product of three to the negative one fifth times three to the positive two fifths. This is going to be equal to, we can take our base three, is gonna be three to the negative one fifth plus two fifths power, plus two fifths power. If you have the same base, the product of that base raised to one exponent and that same base raised to another exponent, that's the same thing as that base raised to the sum of those exponents, a classic exponent property. I encourage you to go on Khan Academy if this is looking foreign or if you need some review. But now this is pretty straightforward. This is going to be equal to, this is is going to be equal to three to the negative one fifth plus two fifths is just one fifth. And there you have it: three to the one fifth power.