Evaluating fractional exponents

We've already seen how to think about something like 64 to the 1/3 power. We saw that this is the exact same thing as taking the cube root of 64. And because we know that 4 times 4 times 4, or 4 to the third power, is equal to 64, if we're looking for the cube root of 64, we're looking for a number that that number times that number times that same number is going to be equal to 64. Well, we know that number is 4, so this thing right over here is going to be 4. Now we're going to think of slightly more complex fractional exponents. The one we see here has a 1 in the numerator. Now we're going to see something different. So what I want to do is think about what 64 to the 2/3 power is. And here I'm going to use a property of exponents that we'll study more later on. But this property of exponents is the idea that-- let's say with a simple number-- if I raise something to the third power and then I were to raise that to, say, the fourth power, this is going to be the same thing as raising it to the 2 to the 3 times 4 power, or 2 to the 12th power, which you could also write as raising it to the fourth power and then the third power. All this is saying is, if I raise something to a power and then raise that whole thing to a power, it's the same thing as multiplying the two exponents. This is the same thing as 2 to the 12th. So we could use that property here to say, well, 2/3 is the same thing as 1/3 times 2. So we could go in the other direction. We could say, hey look, well this is going to be the same thing as 64 to the 1/3 power and then that thing squared. Notice, I'm raising something to a power and then raising that to a power. If I were to multiply these two things, I would get 64 to the 2/3 power. Now, why did I do this? Well, we already know what 64 to the 1/3 power is. We just calculated it. That's equal to 4. So we could say that-- and I'll write it in that same yellow color-- this is equal to 4 squared, which is equal to 16. So 64 to the 2/3 power is equal to 16. The way I think of it, let me find the cube root of 64, which is 4. And then let me square it. And that is going to get me to 16. Now I'll give you in even hairier problem. And I encourage you to try this one on your own before I work through it. So we're going to work with 8/27. And we're going to raise this thing to the-- and I'll try to color code it-- negative 2 over 3 power, to the negative 2/3 power. I encourage you to pause and try this on your own. Well the first thing I do whenever I see a negative exponent is to say, well, how can I get rid of that negative exponent? And I just remind myself, well, the negative exponent really just says, take the reciprocal of this to the positive exponent. I'm using a different color. I'm going to use that light mauve color. So this is going to be equal to 27/8. I just took the reciprocal of this right over here. It's equal to 27/8 to the positive 2/3 power. So notice, all I did, I got rid of the exponent and took the reciprocal of the base right over here. 8/27 is the base. Negative 2/3 is the exponent. Now, how can we handle this? Well, we've already seen that if I have a numerator to some power over a denominator to some power-- and this is another very powerful exponent property-- this is going to be the exact same thing as raising 27 to the 2/3 power-- to the 2 over 3 power-- over 8 to the 2/3 power. This is another very powerful exponent property. Notice, if I have something divided by something and I'm raising the whole thing to a power, I can essentially raise the numerator to that power over the denominator raised to that power. Now, let's think about what this is. Well just like we saw before, this is going to be the same thing. This is going to be the same thing as 27 to the 1/3 power and then that squared because 1/3 times 2 is 2/3. So I'm going to raise 27 to the 1/3 power and then square whatever that is. All this color coding is making this have to switch a lot of colors. This is going to be over 8 to the 1/3 power. And then that's going to be raised to the second power. Same thing we were doing in the denominator-- we raise 8 to the 1/3 and then square that. So what's this going to be? Well, 27 to the 1/3 power is the cube root of 27. It's some number-- that number times that same number times that same number is going to be equal to 27. Well, it might jump out at you already that 3 to the third is equal to 27 or that 27 to the 1/3 is equal to 3. So the numerator, we're going to end up with 3 squared. And then in the denominator, we are going to end up with-- well, what's 8 to the 1/3 power? Well, 2 times 2 times 2 is 8. So this is 8 to the 1/3 third is 2. Let me do that same orange color. 8 to the 1/3 is 2, and then we're going to square that. So this is going to simplify to 3 squared over 2 squared, which is just going to be equal to 9/4. So if you just break it down step by step, it actually is not too daunting.