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Fractional Exponents and Radicals

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Today we're going to talk about fractional exponents and radicals. We already know that an exponent tells us how many times we are multiplying by some number. So how could we multiply by something a fractional number of times?

So today we're going to look at the relationship between a non-fractional exponent like an integer and a radical-- how those are inverse of each other. And that will lead us to the relationship between a fractional exponent and a radical.

So let's start by looking at how a radical and a regular exponent like an integer are related to each other.

So let's say I had 3 to the fifth power. I can cancel that 5 exponent out using a fifth radical. So this 5 exponent and this radical-- the fifth root-- cancel each other out. And so this is just going to equal 3. And that works for any integer exponents and a radical.

So if I had 4 to the third power, I could cancel that out with the cubed root. These two things cancel each other out, so the cubed root of 4 to the third power is just equal to 4. So using this relationship, we can look and see how our fractional exponents can be written as radicals.

So let's look at the relationship between fractional exponents and radicals. If I've got some base a to the fractional exponent m over n, I can say that that's equal, or the same as, the nth root of the same base a to the exponent m.

So using an example with numbers, if I have 2 the 6/7 exponent, I can rewrite that where the denominator of my fraction becomes the index of my radical. And underneath my radical, I have 2 to the numerator of my fractional exponent, so 2 to the sixth.

So let's look at the property we just talked about where the index of a radical and the same exponent will cancel each other out. We looked at the example, the cubed root of 4 to the third, and we talked about the fact that these two things cancel each other out. And this would be just equal to 4.

If we look at this example again using our property, we can also see that that's true because I would rewrite this as my same base 4, the numerator of my fractional exponent is going to be the exponent that is underneath the radical, so 3. And the denominator of my fractional exponent is going to be the index of my radical, so again 3.

Now 4 to the 3/3 is just going to be 4 to the first power. And anything to the first power is just going to be that same base. So here I know that 4 to the first power is just going to be equal to 4.

So again, I see that when I have the index of a radical that matches the exponent underneath the radical, they cancel each other out. And that will evaluate to just my base.

So let's look at a few quick examples again using our property between fractional exponents and radicals. If you are feeling confident, as usual go ahead and pause and then check back with us and see if you got the right answers.

So if i have got the fourth root of 10 to the third power, using our property, I know that my base is going to stay 10. The exponent underneath the radical will become the numerator of my fractional exponent. And the index of the radical will become the denominator of my fractional exponent. So this becomes 10 to the 3/4 power.

Same thing here. My base is going to stay a 9. Exponent underneath the radical becomes my numerator, and the index of the radical becomes my denominator. So this will be 9 to the 8/5.

Let's see what it looks like going the other way. So 4 to the exponent 11/2-- I know underneath my radical I've got my same base 4. The numerator of my fraction will come underneath the radical and stay with my base. And the denominator of my fraction becomes the index of my radical.

Now, a 2 in the index of the radical is just a square root. And generally we do not put the 2 when we're talking about-- generally we don't put the index when we are just talking about a squared root. So that's another good thing to remember, that whenever you don't see a number here that implies a 2, or a square root.

One last example, if I have 12 to the 5/7 exponent, I know underneath my radical I'm going to have 12. My exponent is going to be a 5 underneath the radical. And the index of my radical is going to be the denominator of the fraction, so 7.

So let's go over our key points from today. Exponents and radicals are inverse operations of each other, meaning they cancel each other out. And we also saw that fractional exponents can be written as radicals using the property of some base a written to the fractional exponent m over n is the same or equal as the nth root of the same base a to the exponent m.

So I hope that these key points and examples helped you understand a little bit more about fractional exponents and radicals. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.

Formulas to Know

- Property of Fractional Exponents