Rational Exponents: Study with Video Lessons, Practice Problems & Examples

Understanding radicals and their relationship with exponents is crucial in mathematics. A radical expression, such as the square root of a number, can be rewritten using rational exponents. For instance, the square root of 5 is equivalent to \(5^{1/2}\). The general rule is that for a term \(a\) raised to the power \(m\) with an index \(n\), it can be expressed as \(a^{m/n}\). This conversion aids in simplifying expressions and solving equations effectively, enhancing comprehension of polynomial terms and their properties.

Rational Exponents

Welcome back, everyone. So we've talked a lot about radicals and how they are very related to exponents. For example, if I take the square root of a number, that's the opposite of squaring a number, and so on. What I'm going to show you in this video is you can actually take a radical expression like the square root of 5, and we can actually rewrite that as an exponent. To do that, we're going to use these things called rational exponents. Alright? Let me go ahead and show you how this works. We can rewrite a radical expression as a term with an exponent that is a fraction. That's why these things are called rational or sometimes called fractional exponents. For example, if I have the square root of 5 squared, then what I know from square roots is that the square root of a square basically just undoes it, and then you just get 5. Right? So we've seen that before. Now let's say I have something like 5 to the one-half power. Now I've never seen that before, but, basically, just bear with me here. But we do know that if you take 5 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 2 just equals, well, 1. So in other words, this just becomes 5 to the 1 power. So in other words, when I took the radical if I took the square root of 5 and squared that, I just got 5. And if I take 5 to the one-half power and I square that, I also get 5. So, basically, these two things just mean the exact same thing. The square root of 5, 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. Alright? Now the general way that you're going to do this, and I know this looks a little bit scary at first, is you can basically just take an index and a power of a term, and you can just convert that into a fractional exponent, where the top is the power of the thing that's inside the radical, and the bottom, the denominator, is going to be the index or the root. For example, we said 5 to the one-half power is equal to radical 5, and that's because what happens is there's an invisible one that's here inside of this 5 that's inside the radical, and the 2 is actually the index of the square roots, which is also kind of invisible. Right? So in other words, 5 to the one-half power is just 5 inside the radical, and that whole thing is square rooted over here. That's the whole thing. Alright? So let's go ahead and get some practice here of converting radicals to rational exponents. Let's take a look. So we're going to rewrite radicals as exponents, or we're going to do the opposite, rewrite exponents as radicals. Let's take a look at the first one here, 13 to the one-over-3 power. So I have a term here, and I've got a fractional exponent. Remember, the bottom is going to 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 3, so that goes over here. And then I just get 13, and the one basically just goes in here inside and is 13 to the one power. That's how you convert a fractional exponent into a radical. Now we're going to do the opposite here. Now we're going to take something like square root of x, and we're going to convert that to a fractional exponent. So how do we do this? Well, basically when we did this for radical 5 or square root of 5, the square root of 5 just became 5 to the one-half power. We can just do the exact same thing with variables. So in other words, this just becomes x to the one-half power. Alright? So that is the answer. Alright. So now let's go ahead and do this a little bit more complicated expression over here in case c. So here we have an index or root of 5, and here we have a term that's raised to the second power over here. So how does this go? Well, remember, what happens is the index is going to be the denominator of your fraction, and the power of the term inside the radical is going to be the top. So in other words, when I convert this, what ends up happening is I just get y, and this 2 is at the top, and it's divided by 5, which is on the bottom. So that's how you do that. Alright? And that's how you convert them. Alright. So that's it for this one, folks. Let me know if you have any questions.