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 happens then? So let's explore this. For example, what would it mean to say 2 to the power of 1/2? Okay, let's think about this, we can 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 1/2 = K. Well, notice that if we multiply 1/2 by 2, we get a whole number, and of course, we could use that multiplying exponent rule if we raise 2 to the 1/2 to another power. So I'm gonna say, why don't we square both sides. And on one one side, we get K-squared. On the other side, we get 2 to the 1/2 squared.
And of course, the laws of exponents say, we multiply those exponents 1/2 times 2 = 1. So that side just becomes ordinary 2, K-squared = 2. But of course, we can solve this K very easily. Take a square root, K equals the square root of 2, and that must be what K equal.
So in other words, 2 to the 1/2 equal the square root of 2, raising something to the power of 1/2 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 1/3? Well, you might guess but we'll follow the same process. Again, fill in the dummy variable K.
And now, notice that if we multiply 1/3 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 1/3 to the 3, while the 1/3 and the 3 get multiplied and that just equals 1, so it's 2 to the 1 or ordinary 2. So K-cubed = 2.
We can take the cube root of both sides, K = the cube root of 2. So in other words, 2 to the 1/3 equals the cube root of 2. Well, you might see a general pattern emerging here. In other words, if we take something to the 1/2, it's the square root. If we take something to the 1/3, it's the cube root. You might guess, if we take it to the 1/4, it's the 4th root, 1/5, 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 explicit link between fractional exponents and roots. So for example, if we had something like 6 to the power of 1/7, 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 six. 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 3/5? Well, remember, we can write 3/5 as either 3 times 1/5 or 1/5 times 3.
Of course, we can write the product either way, and this has implications of the laws of exponents. I can write it as 3 times 1/5, have the power of 3 inside and have the 1/5 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. Eight to the 4/3. Well, we could write that either as the cube root of 8 to the 4th or 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 because 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 cubed root of it, hmm, that sounds doubtful. Whereas, 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 the 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 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 of zero, although the roots of zero 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 1/2. 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.
Read full transcriptWhat happens then? So let's explore this. For example, what would it mean to say 2 to the power of 1/2? Okay, let's think about this, we can 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 1/2 = K. Well, notice that if we multiply 1/2 by 2, we get a whole number, and of course, we could use that multiplying exponent rule if we raise 2 to the 1/2 to another power. So I'm gonna say, why don't we square both sides. And on one one side, we get K-squared. On the other side, we get 2 to the 1/2 squared.
And of course, the laws of exponents say, we multiply those exponents 1/2 times 2 = 1. So that side just becomes ordinary 2, K-squared = 2. But of course, we can solve this K very easily. Take a square root, K equals the square root of 2, and that must be what K equal.
So in other words, 2 to the 1/2 equal the square root of 2, raising something to the power of 1/2 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 1/3? Well, you might guess but we'll follow the same process. Again, fill in the dummy variable K.
And now, notice that if we multiply 1/3 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 1/3 to the 3, while the 1/3 and the 3 get multiplied and that just equals 1, so it's 2 to the 1 or ordinary 2. So K-cubed = 2.
We can take the cube root of both sides, K = the cube root of 2. So in other words, 2 to the 1/3 equals the cube root of 2. Well, you might see a general pattern emerging here. In other words, if we take something to the 1/2, it's the square root. If we take something to the 1/3, it's the cube root. You might guess, if we take it to the 1/4, it's the 4th root, 1/5, 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 explicit link between fractional exponents and roots. So for example, if we had something like 6 to the power of 1/7, 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 six. 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 3/5? Well, remember, we can write 3/5 as either 3 times 1/5 or 1/5 times 3.
Of course, we can write the product either way, and this has implications of the laws of exponents. I can write it as 3 times 1/5, have the power of 3 inside and have the 1/5 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. Eight to the 4/3. Well, we could write that either as the cube root of 8 to the 4th or 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 because 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 cubed root of it, hmm, that sounds doubtful. Whereas, 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 the 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 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 of zero, although the roots of zero 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 1/2. 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.
