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? Read full transcript
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 would 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 could solve this for k very easily. Take a 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, equaled 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 1. Now, what would it mean to say 2 to the 1 3rd?
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 1 3rd times three, 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 1 3rd to the 3 well the 1 3rd and the 3 get multiplied and that just equals 1 so it's 2 to the 1 or ordinary 2.
So k cubed equals 2. We can take the cube root of both sides. K equals the cube root of two. So in other words, two to the one third equals the cube root of two. 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, 1 3rd, its the cubed root. You might guess, if we take it to the one 4th, it's the 4th root, one 5th, it's the fifth 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 explicit link between fractional exponents and roots.
So for example, if we had something like 6 to the power of 1, 7th, what that would mean is the 7th root of 6. What exactly does that mean, the 7th 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 1 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 3 5ths. Well, remember, we can rank 3 5ths as either 3 times 1 5th or 1 5th times 3, we can write, of course we can write the product either way and this has implications with the laws of exponents. I can write it as 3 times 1 5th, have the power of 3 inside, and have the 1 5th 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 4 3rds.
Well, we could write that either as the cube 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 e would have to do is figure out 8 to the power of 4.
Well that's going to be a large number. 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 cubed root of it. Hm, that sounds doubtful.
Where as with the other one, all we have to do is take a cubed root of 8, we can 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 cubed root of 8 is just 2. So, we get 2 to the 4th, and that's 16. So, these rules that we talked about. Are rules that are true for positive numbers, when b is a positive number.
Technically they are true at 0, though the roots of 0 are an unlikely topic on the test. If the denominator of the exponent 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 cube root, or the 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.