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Fractional Exponents

Fractional exponents. In this lesson, we can make explicit the link between roots and exponents. So far, the only exponents we have considered have been integers, either positive, negative, or zero. So we’ve been sticking with integers. What happens if the exponent is not an integer, but a fraction?

What, what happens then? So let’s explore this. For example, what would it mean to say 2 to the power of one-half? Well, gee, let’s think about this. We could do mathematical operations if we have another side to that equation. So just let’s create a dummy variable K.

We’ll call the output K. 2 to the one-half equals K. Well, notice that if we multiply one-half by 2, we get a whole number. And, of course, we could use that multiplying exponent rule if we raise 2 to the one-half to another power. So I’m gonna say, why don’t we square both sides?

Well, then one side we get K squared. On the other side, we get 2 to the one-half squared. And, of course, the laws of exponents say we multiply those exponents. One-half times 2 equals 1. So that side just becomes ordinary 2. K squared equals 2.

Well, of course we can solve this for K very easily. Take the square root. K equals the square root of 2. And that must be what K equals. So in other words, 2 to the one-half equals the square root of 2. Raising something to the power of one-half is the same as finding the positive square root of it.

That’s important fact number one. Now, what would it mean to say 2 to the one-third? Well, you might guess, but we’ll follow the same process. Again, fill in a dummy variable K. And now, notice that if we multiply one-third times 3, we’ll get a whole number.

So, we’ll cube both sides and, of course, the right side just becomes K cubed. The left side, 2 to the one-third to the 3. Well, the one-third and the 3 get multiplied and that just equals 1, so it’s 2 to the 1 or just ordinary 2, so k cubed equals 2. We can take the cube root of both sides. K equals the cube root of 2.

So in other words, 2 to the one-third equals the cube root of 2. Well, you might see a general pattern emerging here. In other words, if we take something to the one-half, it’s the square root. If we take something to the one-third, it’s the cube root. You might guess if we take it to the one-fourth, it’s the 4th root, one-fifth, it’s the 5th root, that sort of thing.

And, in fact, we can generalize by saying b to the power of 1 over m is the mth root of b. So this is the explicitly link between fractional exponents and roots. So for example, if we had something like 6 to the power of one-seventh, what that would mean is the 7th root of 6. What exactly does that mean, the seventh root of 6?

This is the number which, when raised to the 7th power, equals 6. What happens if the exponent is a fraction that has a number other than one in the numerator? So far, we’ve been looking only at fractions that have one in the numerator. What would happen if we had something like 2 to the three-fifths? Well, remember we can write three-fifths as either 3 times one-fifth or one-fifth times 3.

We can write, of course we can write the product either way, and this has implications for the laws of exponents. I can write it as 3 times one fifth, have the power of 3 inside and have the one-fifth outside. And so, that would be the 5th root of 2 cubed, or the 5th root of 8. That would be one way I could do it.

Another way I could do it would be to write the 5th on the inside. So on the inside, I have just the 5th root of ordinary 2, and on the outside, I’m cubing it. Either one of those is perfectly fine. And I will say if you actually have to do a calculation, if you have to actually choose between these two, always make things smaller before you make things bigger.

That’s a very important point of strategy. Here’s a practice problem where you can apply some of this. Pause the video and then we’ll talk about this. Okay, 8 to the four-thirds. Well, we could write that either as the cubed root of 8 to the 4th, or the cube root of 8, that whole thing to the power of 4.

We could write it either way. Now, the question is, which would be a better way to calculate? Well, with that first one, the first thing we’d have to do is figure out 8 to the power of 4. Well, that’s gonna be a large number. Of course, 8 to the power of 4 is gonna be 8 squared squared, so that would be 64 squared.

I don’t know 64 squared off the top of my head, but that’s gonna be a very large number, and then we’re gonna try and take a, a cube root of it? Hm, that’s sounds doubtful. Whereas with the other one, all we have to do is take a cube root of 8. We could do that and then raise it to the 4th. So that first one is just a horrible idea.

Don’t raise it to some high power and then try to find a root. Find the root first. That’s an enormous point of strategy. So, we’ll find the root first, and of course, the cube root of 8 is just 2, so we get two to the 4th and that’s 16. So these rules that we’ve talked about are rules that are true for positive numbers when b is a positive number.

Technically, they are true of zero though the roots of zero are an unlikely topic on the test. If the denominator of the expoent-fraction is odd, then the base can be negative as well. Remember that we could not take even roots of negative numbers, but we could take odd roots of negative numbers.

For example, the cubed root, or a 5th root of a negative number. In summary, roots are represented by fractional exponents. That’s the big idea. The square root of a quantity equals that quantity to the power of one-half. That is by far the most common fractional exponent you’ll see on the exam. The power b to the 1 over n means the nth root of b.

And the power b to the m over n can be written either as the root of the power or as the root to the exponent m.