Rational Exponents - Online Tutor, Practice Problems & Exam Prep

Radicals can be expressed as rational exponents, where the square root of a number, such as $5^{\frac{1}{2}}$, is equivalent to $\sqrt{5}$. The general form for converting is $a^{\frac{m}{n}}$, where m is the power and n is the index. This understanding aids in simplifying expressions and solving equations efficiently.

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. And to do that, we're going to use these things called rational exponents. Alright? Let me go ahead and show you how this works. So 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 \(5^{\frac{1}{2}}\). Now, I've never seen that before and, basically, just bear with me here. But we do know that if you take \(5^{\frac{1}{2}}\) 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 \(\frac{1}{2} \times 2\) just equals, well, 1. So, in other words, this just becomes \(5^{1}\). 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 \(5^{\frac{1}{2}}\) and 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 \(5^{\frac{1}{2}}\) is equal to square root of 5, and that's because what happens is there's an invisible one that's here, inside of this 5 that's in the radical, and the 2 is actually the index of the square roots, which is also kind of invisible. Right? So, in other words, \(5^{\frac{1}{2}}\) 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 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, \(13^{\frac{1}{3}}\). 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, so that goes over here. And then I just get 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 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 square root of 5, square root of 5 just became \(5^{\frac{1}{2}}\). We can just do the exact same thing with variables. So, in other words, this just becomes, sorry, \(x^{\frac{1}{2}}\). 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 part 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.

Here’s what students ask on this topic:

How do you convert a radical expression to a rational exponent?

To convert a radical expression to a rational exponent, use the general form $a^{\frac{m}{n}}$, where $m$ is the power and $n$ is the index (root). For example, the square root of 5 can be written as $5^{\frac{1}{2}}$. Here, the index is 2 (square root), and the power is 1. This method helps in simplifying expressions and solving equations efficiently.

What is the rational exponent form of the cube root of 8?

The cube root of 8 can be expressed as a rational exponent. The general form is $a^{\frac{m}{n}}$, where $m$ is the power and $n$ is the index. For the cube root of 8, the index is 3 and the power is 1. Therefore, it can be written as $8^{\frac{1}{3}}$.

How do you simplify expressions with rational exponents?

To simplify expressions with rational exponents, use the properties of exponents. For example, $a^{\frac{m}{n}}$ can be simplified by converting it back to a radical form if needed. Additionally, use the power rule ${\mathrm{(a^m)}}^n$ = $a^{\mathrm{mn}}$ and the product rule ${\mathrm{(ab)}}^m$ = $a^m{b}^m$ to combine and simplify terms.

What is the relationship between radicals and rational exponents?

Radicals and rational exponents are two ways to represent the same mathematical concept. A radical expression like the square root of a number can be written as a rational exponent. For example, the square root of 5 is equivalent to $5^{\frac{1}{2}}$. The general form is $a^{\frac{m}{n}}$, where $m$ is the power and $n$ is the index (root). This relationship helps in simplifying and solving equations more efficiently.

How do you convert a rational exponent to a radical expression?

To convert a rational exponent to a radical expression, use the general form $a^{\frac{m}{n}}$, where $m$ is the power and $n$ is the index (root). For example, ${13}^{\frac{1}{3}}$ can be written as the cube root of 13. Here, the index is 3, and the power is 1. This method helps in visualizing and simplifying expressions.