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Simplifying rational exponent expressions: mixed exponents and radicals

Video transcript

- [Voiceover] So I have an interesting equation here. It says V to the negative six fifths power times the fifth root of V is equal to V to the K power, for V being greater than/equal to zero. And what I wanna do is try to figure out what K needs to be. So what is... what is K going to be equal to? So pause the video and see if you can figure out K, and I'll give you a hint, you just have to leverage some of your exponent properties. Alright, let's work this out together. So the first thing I'd want to do is being a little bit consistent in how I write my exponents. So here I've written it as negative six fifths power, and here I've written it as a fifth root, but we know that the fifth root of something... we know that the fifth root... the fifth root of V, that's the same way, that's the same thing as saying V to the one fifth power, and the reason I want to say that is because then I'm multiplying two different powers of the same base, two different powers of V. And so we can use our exponent properties there. So, this is gonna be the same thing as V to the negative six fifths times, instead of saying the fifth root of V I can say times V to the one fifth power is going to be equal to V to the K. It's gonna be equal to V to the K power. Now, if I'm multiplying V to some power times V to some other power, we know what the exponent properties would tell us, and I could remind us. I'll do it over here. If I have X to the A times X to the B, that's going to be X to the A plus B power. So here, I have the same base, V. So this is going to be V to the, and I could just add the exponents. V to the negative six fifths power plus one fifth power, or V to the negative six fifths plus one fifth power is going to be equal to V to the K. Is equal to V to the K. I think you might see where this is all going now. So this is going to be equal to V. So negative six fifths plus one fifth is going to be negative five fifths or negative one. So all of this is going to be equal to negative one, and that's going to be equal to V to the K. So K must be equal to negative one, and we're done. K is equal to negative one.