Simplifying Expressions Containing Rational Exponents, Part 1 070 - 16a

This video is going to be about simplifying expressions with rational exponents. So let's look at an example. Here I've got an expression, x-to-the-1/4 times 8x-to-the-1/2 and that 8x-to-the-1/2 is raised to the negative 2/3. Now, the thing to remember about dealing with something like this is we're going to use exactly the same rules for the exponents that we would use if the exponents were whole numbers. Okay, so I guess the place to start would be to take this 8x-to-the-1/2 and raise everything to the negative 2/3. If this is confusing to you, ask yourself what you would do if it was 8x-squared raised to the negative 2. well you would take the negative 2 and multiply it by the exponents that you have in parentheses. So we're going to do that, but we're going to do it with the fractions. So let me copy down my x-to-the-1/4. I'm not ready to deal with that yet. And I'm gonna take this 8 and call it 8-to-the-first, just so we can remember that it has an exponent. So now I want to take 8-t0-the-1st-power and raise that to the the negative 2/3 power. So I've got to multiply the one by negative 2/3. So that will give me 8- to-the- negative-2/3. And then I've got x-to-the-1/2, and I've got to raise that to the negative 2/3. So I'm going to have to multiply 1/2 times negative 2/3. So let's see how we do that. 1/2 times negative 2/3. Well I've got a 2 in the denominator, and a 2 in the numerator, so I can cross those out. I want to keep the negative sign, though. So I'll cross out this 2, it's gonna be a 1. And this negative 2 will be a negative 1. And if I'm multiplying fractions, I multiply across the numerators. So that's going to be 1 times negative 1 is negative 1. And 1 times 3 is 3. So that means I'm going to have x-to-the negative- 1/3. Okay. Let's see what the next step is. Now I guess it's time for me to multiply x-to-the-1/4 times this expression, 8-to-the-negative-2/3 x-to-the-1/3. So I'm going to have... Again, if you get confused about this ask yourself what you would do if these were whole number exponents, if it was just, let's say, x-to-the-4th times 8x-to-the-3rd. Well, we wouldn't touch the 8, so I'm not gonna touch the 8, I'm just gonna write down 8-to-the-negative-2/3. Then I have to multiply x-to-the-1/4 times x-to-the-negative-1/3. So I know I'm gonna have an x. If I multiply two similar bases, an x and an x, and we have exponents, I have to add the exponents together. So that means I have to add 1/4 plus negative 1/3. This might be a good time to remind you that if you need to do a fraction review you should probably do the fraction review. At any rate, If I'm adding fractions and the denominators are not the same, I have to find a common denominator. The common denominator is going to be 12. So I want to turn this 4, from the 1/4, into a 12. I can do that by multiplying it by 3. But I can't just multiply by 3. I have to multiply the whole fraction by the fraction 3/3, because all I'm doing then is just multiplying by 1, and I haven't changed it. Make sure this makes sense to you. Review your fractions if you need to and think about this. Then I'm going over to the other faction. I've got this negative 1/3. I want to change its denominator to 12. So I'm going to have to multiply that 3, in the denominator, by a 4. Well, if I multiply the denominator by 4, I've got to multiply the numerator by 4. So the denominators are going to become 12. I know that. And I'm going to have 3 times 1 is 3, and negative 1 times 4 is negative 4, so I'm going to have 3 minus 4. And 3 minus 4 is negative 1. So that means I'm going to have x- to-the-negative- 1/12. Okay? So it's simplified quite a bit. I've got 8-to-the-negative-2/3 times x-to-the-negative-1/12. Since both of these have negative exponents, negative exponents are going to become positive if I get them down into the denominator. So I'm going to make a fraction. One will be the whole numerator. And I'm going to have 8-to-the-2/3 times x-to-the-1/12. Now the only thing left to do is, we don't want to leave that 8-to-the-2/3 like it is. So let's figure out what that would be. I know you can do this in your calculator, but let's just do it on paper so you understand the process. Well, 8-to-the-2/3 is the same as the 3rd root of 8 squared, and I can write this square either after the 8, but I'm gonna write it after the whole expression, because it's gonna make it easier. So I'm gonna write the 3rd root of 8, and that whole thing is going to be squared. So now I ask myself what the 3rd root of 8 is. Well, the 3rd root of 8 is 2, because 2 times 2 times 2 is 8. So that means this 3rd root of 8 is going to be 2, and I want to square that, and 2-squared is 4. So now instead of having this one over 8-to-the-2/3 x-to-the-1/12, I'm going to simplify that down to one over 4 x- to-the-1/12. And that's as far as I can go with it. Okay? so I want to do a second one, but these are kind of lengthy, so I'll do it in a second video. Stick around. I'll be back soon.