 Let's talk about the radicals and see how we can combine things that are a little more complicated from all these operations that we've learned. So for example, let's do, let's say you have to square root of 8 times 32 to the power of negative 3 over 4, so we've got square root of 8, is that legible, is that legible? Square root of 8 times 32 to the power of 3 over 4. Now these are different bases, 8 and 32 are different bases, so we can't directly get the exponents involved. The only way we can get the exponents directly involved with this multiplication, add the exponents together, is if we have the same base. Now what we need to do is convert the 8 and the 32 to the same base, and I made up the question, and I know they go down to the same base, but if you were getting this for the first time, and if you didn't know they were going down to the same base, the only way you can do it is break it down to its prime factors, right? So let's break these down to their prime tree, use the prime tree, break them down to their prime factor, 32 breaks down into 2 times 2 times 2, right? 2 times 2 times 2 is 2 to the power of 3, right? So this guy becomes 2 to the power of 3. So square root of 8 is 2 to the power, or, forgot about the square root symbol. The square root symbol, as we talked about before, the 2 over here goes to the denominators of power. So square root of 8 is 2 to the power of 3 over 2, 8 is 2 to the power of 3. Square root of 8 is 2 to the power of 3 over 2. Now let's look at 32, 32 we're going to break down to its prime numbers, right? So 32 we have 4 times 8, and 4 is just 2 times 2, 8 is 2 times 2 times 2 times 2, 2, 2, 2. So 8 is not 32, it's 2 to the power of 5. So the way this works is it becomes 2 to the power of 5. 2 to the power of, we've got that guy up there too, right? 2 to the power of negative 3 over 4. Now what we're going to do is apply our power rule, power to a power, means multiplication, right? So this becomes 5 over 1, so with multiplication, with fractions, bottom, bottom, bottom. This thing is the same, 2 to the power of 3 over 2, and by the way, this would have been 2 to the power of 3 to a power of a half, which would have been 3 times a half, is 3 over 2. I just skipped the step there, but I'm showing it over here, so hopefully you can do it over here too, right? So 5 over 1 times 3 over 4, 5 times 3 is 15, 1 times 4 is 4. So this becomes 2 to the power of negative 3 to the power of 4. Now we have them as the same base. Now that we've got them as the same base, we can add the exponents. So let's go over here. We got, I'm going to do it with this so it doesn't get interfere with this, okay? So we take it up here. This is, the base is the same, 2, 3 over 2, multiplication means addition, yes. And those are all in the exponent, okay? So this will be 2, the common denominator for 2 and 4 is 4. We multiply 2 by 2, we multiply it 3 by 2, so this becomes 6 minus 15. So the answer to this is 6 minus 15 is 9, negative 9, so it's negative 9 over 4. So the answer to the original question, which was once the square root of 8 times 32 to the power of negative 3 over 4 is 2 to the power of negative 4 over 2. Now sometimes in the questions they give you in school or doing a test, they'll say they want it to be positive powers only. So if you want to convert this to a positive power only, the negative power here doesn't flip this. Some people make the mistake of flipping this. This is an operation on the base number, so it flips this. So the answer to this would be, an alternate answer to this would be 1 over 2 to the power of, that's another way of expressing it. Some people will want it as a negative power, some people will want it as a positive power. Now this should be a 9, not an 8. And sometimes they'll say use radicals, no fractions in the exponents. Now the other way you could express the same question. Same thing is, take this 4, put it in the root, in front of the root. So it would be, this is equal to 4 through the, that's supposed to be 4 there, 4 through the 2 to the power of negative 9. Again, you could take another step further and say no negative power, so it would be 4 through the 1 over 2 to the power of 9. Or you could take the 9 and expand it further. So that's one way, actually I'm going to show you another answer to this. Let's break this down into its radical form. And what we're going to do with the negative 1 is we're going to kick it out at the end, because I don't want to deal with fractions throughout the whole thing. I don't want to do the fraction at the end. So the way you could express this is the 4 through the 2 to the power of 9, all to the negative 1. This expression is the same as that. Because normally in the exponent, in the denominator, when you've got an exponent, the denominator kicks up to the front. And make sure you put the 4 there because if you don't it just means the square root. Now the way this works is this is equivalent to the 4 through the 1 and 9, 2 to the power of 9 means 2 multiplied by itself, 9 times. 1, 2, 3, 4, 5, 6, 7, 8, 9. So all those 2s multiplied together means 2 to the power of 9. Now what the 4 means here is you're looking for quadruples. You're looking for 4 things, combining 4 of them together so you can bring them out as a single. So these 4 guys, you've got 4 guys here. So these 4 guys can come out of this boundary as a single 2. These 4 guys can come out of the same boundary as a single 2. So you've got 2 coming out and another 2 coming out. When they come out they multiply each other, so 2 times 2 is 4. So this will be 4, 4 through the 1. What do you got left in here? You've got another 2 left in here. You've got just one 2 left in here, right? So you put a 2 there to the power of negative 1. Now this might be sufficient or they might say expresses it a positive power and the way you would express it is a positive power. Let's see if we can go here. Let's write it down here. The way you would express it as a positive power is kick this whole thing down. And the way you kick it down is the same answer to this would be, are we on the board? We're on the board? Would be 4, 4 through the 1. Yeah, because negative 1 kicks it down. So that expression and that expression are identical. And they're the same as the other 2 expressions. It's just the question of, you know, what's the cleanest way that you want to present it? Or how the solution is in the, what the answer is in the multiple choices that they gave you for some multiple choice exam or what your teacher prefers. Or, you know, according to the question, they might tell you they don't want any negative powers or they don't want any radicals. You know, they just want it as an exponent or they do want it as a radicals. No fractions in the exponent. So it really depends what specifically they're looking for and how the question is phrased for you to decide which way you're going to give the answer. Personally, if it was me, that's good enough for me. It's the cleanest one I could see.