Hi, my name is Anthony Varela. And today, I'm going to talk about fractional exponents and radicals. So we're going to talk about exponents and radicals as operations. Then we'll relate radicals to fractional exponents. And then we're going to practice converting between radicals and expressions with fractional exponents.
So, to get started, I'm going to talk about exponents and radicals as inverse operations. So let's take a positive number, 4, and we're going to square this number. Then we're going to take the square root of 4 squared. This will bring me back to 4.
Let's take a look at another example. How about 6? And I'm going to cube this number. And now I'm going to take the cubed root of 6 cubed. That will bring me back to 6. So we can say that exponents and radicals are inverse operations. And, more specifically, the nth roots of a number to the nth power equals that number, so long as we're talking about non-negatives.
All right. So now what if the power and the index are not the same? So here's an example of that. The cubed root of 2 to the fifth power. Our power and our index are not the same. So it doesn't fit this rule. I know it does not equal 2.
Well, I'm going to take what's underneath our radical, 2 the fifth. And to show that I'm taking the cubed root of this, I can use a fractional exponent. And that's the big idea for today's lesson. I'm going to raise all of that to the 1/3 power.
And now to clean this up a bit, I can multiply 5 by 1/3 due to our power of power property of exponents. So the cubed root of 2 to the fifth power equals 2 raised to the 5/3. So now let's compare then our expression with our fractional exponent to our radical expression here.
And I notice that the 5 corresponds to the-- it's numerator in my fraction. And it's also the power underneath the radical. And the 3 is the denominator of this fractional exponent. It's also the index of the radical. It tells me what type of root I'm taking.
So radicals can be expressed using fractional exponents. So if we have a positive number a. We raise that to the power of m. And then take the nth root, I can rewrite this as our base a raised to a fraction, a fractional exponent, m in the numerator, and n in the denominator. So we're going to write that down.
All right. So now I'd like to very briefly connect this to the whole idea that exponents and radicals are inverse operations. So we know from before that this equals 7. But I'd like to show this using fractional exponents.
So I'm going to take our cubed root of seven cubed. And I'm going to write this as 7 cubed raised to the 1/3 power. Now I can multiply those two exponents to get 7 to the 3/3 power. 3 over 3 equals 1. And anything raised to the first power remains the same. So there I have shown another way that the cubed root of 7 cubed equals 7.
All right. So now let's practice writing radicals as fractional exponents. So here I have the fifth root of 6 to the fourth power. So how can I get that to look something like this? We have a base raised to a fractional exponent.
So let's take what's underneath our radical, 6 to the fourth power, and now let's raise all of that to a fractional exponent. So it would be 1 over 5 because we're taking the fifth root. And now we can multiply these two exponents together due to our power of a power property. And we have 6 raised to the 4th-- 4/5.
Let's do one more example. Here, we have the cubed root of 3.2 raised to the seventh power. So I'm going to take our 3.2. And now I need to make a fractional exponent using 3 and 7. What's going to be my numerator? And what's going to be my denominator? Well, 7 is going to be the numerator. And 3 is going to be the denominator.
Now let's go in the opposite direction. Let's start with a fractional exponent and write it as a radical. So I have 8 to the 2/3 power. Well, I can think of 2/3 as being the product of two exponents. So I'm going to write this as 8 squared. And then I'm going to take all of that to the 1/3 power. So I'm breaking 2/3 up into 2 times 1/3.
And now I can rewrite this then as 8 squared. And I'm going to take the cubed root of that quantity. So the cubed root of 8 squared is the same thing as 8 to the 2/3 power.
We'll go through another example. Here, I have 0.5 raised to the 5/4. And we'd like to write this using a radical sign. So I'm going to take our base, 0.5, and let's see, my numerator, my fractional exponent, relates to the power of this base. So I'm going to raise that to the power of 5. And I know that the denominator here of my fraction means I'm going to take a radical. But what kind? It's important to write the 4. So that's how we write that then using a radical sign.
So let's review our notes. Today, we talked about exponents and radicals as inverse operations. So nth root of a number to the nth power equals that number, so long as that number is non-negative. And radicals can be written as fractional exponents in the other way around.
So we talked about how this fractional exponent here, our numerator, corresponds to this power here. And our denominator corresponds to the index of the radical or what type of root we're going to take.
So thanks for watching this video on fractional exponents and radicals. Hope to catch you next time.
