Welcome back everyone. So we've talked a lot about radicals and how they were very related to exponents. For example, like if I take the square root of a number, that's the opposite of squaring a number. And so on, what I'm gonna show you in this video is we can actually take a radical expression like the square root of five. And we can actually rewrite that as an exponent. And to do that, we're gonna use these things called rational exponents. All right, let me go ahead and show you how this works. So we can rewrite a radical expression as a term with an exponent that are fractions. So that's why these things are called rational or sometimes called fractional exponents. For example, if I have the square root of five squared, then what I know from square roots is that the square root of a square basically just undo it. Um And then you just get five, right? So we've seen that before. Now, let's say I have something like five to the one half power. Now, I've never seen that before and, and basically just bear with me here. But we do know that if you take five to the one half power and you square that we know how to deal with this by using our rules of exponents. Remember we talked about the power rule where you basically just multiply their exponents and one half times two just equals, well, one. So in other words, this just becomes five to the one power. So in other words, when I took the radical, if I took the square to five and squared that I just got five, and if I take five to the one half power and I square that I also get five. So basically these two things just mean the exact same thing, the square root of five. Another way I can represent that is instead of using radicals, I can use now, fractional exponents. That's the whole thing is that these two things just mean the exact same thing. All right. Now, the general way that you're gonna do this and I know this looks a little bit scary at first is you can basically just take uh a, an index and a power of a term and you can just convert that into a fractional exponents where the top is the power of the thing that's inside the radical and the bottom, the denominator is gonna be the index or the root. For example, we said five to the one half power is equal to radical five. And that's because what happens is there's an invisible one that's here. Um in this inside of this five, that's in the radical. And the two is actually the index of the square roots, which is also kind of invisible, right? So in other words, five to the one half power is just five inside the radical. And that whole thing is square rooted over here. That's the whole thing. All right. So let's go ahead, go ahead and get some practice here of converting radicals to rational exponents. Let's take a look. So we're gonna rewrite radicals as exponents or we're gonna do the opposite, rewrite exponents as radicals. Let's take a look at the first one here, 13 to the 1/3 power. So I have a term here and I've got a fractional exponents. Remember the bottom is gonna be the index or the roots and the top is going to be the thing that's inside of the radical. So when I convert this, what happens is I can write this as a root. What is the root? It's three. So that goes over here. So that goes over here and then it just gets 13 and the one basically just goes in here inside and it's 13 to the one power. That's how you convert a fractional exponent into a radical. Now, we're gonna do the opposite here. Now we're gonna take something like squared of X and we're gonna convert that to a fractional exponents. So how do we do this? Well, basically, when we did this for radical five or square to five square to five just became five to the one half power, we can just do the exact same thing with uh with variables. So in other words, this just becomes uh sorry X to the one half power. All right. So that is the answer. All right. So now let's go ahead and do this a little bit more complicated expression over here in C. So here we have an index or root of five and here we have a term that's raised to the second power over here. So how does this go? Well, I remember what happens is the index is gonna be the denominator of your fraction and the power of the term inside the radical is gonna be the top. So in other words, when I convert this, what ends up happening is I just get Y and this two is at the top and it's divided by five, which is on the bottom. So that's how you do that. All right. And that's how you convert them. All right. So that's it for this one. Folks. Let me know if you have any questions.

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