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Simplifying mixed radical and exponential expressions

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We're asked to simplify r to the 2/3 s to the third, that whole thing squared. Times the square root of 20r to the fourth s to the fifth. Now this looks kind of daunting, but I think if we take it step by step it shouldn't be too bad. So first we can look at this first expression right here where we're taking this product to the second power. We know that instead we can take each of the terms in the product to the second power and then take the product. So this is going to be the same thing as r to the 2/3 squared times s to the third squared. And now let's look at this radical over here. We have the square root, but that's the exact same thing as raising something to the 1/2 power. So this is equal to-- so times this part. Let me do this in a different color. This part right here, that is the same thing as 20. And instead of just writing 20, let me write 20 as the product of a perfect square and a non-perfect square. So 20 is the same thing as 4 times 5. That's the 20 part. Times r to the fourth times s to the fifth. Now let me write s to the fifth also as a product of a perfect square and a non-perfect square. r to the fourth is obviously a perfect square. Its square root is r squared. But let's write s to the fifth in a similar way. So s to the fifth we can rewrite as s to the fourth times s. Right? S to the fourth times s to the first, that is s to the fifth. And of course, all of this has to be raised to the 1/2 power. Now let's simplify this even more. If we're taking something to the 2/3 power and then to the second power, we can just multiply the exponents. So this term right here, we can simplify this as r to the 4/3 power. And just as a bit of review, taking something to the 4/3 power, you can view it as either taking-- finding its cube root, taking it to the 1/3 power, and then taking its cube root to the fourth power. Or you can view it as taking it to the fourth power and then finding the cube root of that. Those are both legitimate ways of something being raised to the 4/3 power. So you have r to the 4/3 times s to the 3 times 2. Times s to the sixth power. And then we could raise each of these terms right here to the 1/2 power. So times-- let me color code it a little bit. And we actually wouldn't need the parentheses once we do that. Times 4 to the 1/2 times 5 to the 1/2. That term right there. Times r to the fourth to the 1/2 power. Times-- I might run out of colors-- s to the fourth to the 1/2 power. We're raising each of these terms to that 1/2 power. Times s to the 1/2 power. There's a lot of ways we can go with this, but the one thing that might jump out is that there are some perfect squares here and we're raising them to the 1/2 power. We're taking their square roots, so let's simplify those. So this 4 to the 1/2, that's the same thing as 2. We're taking the principal root of 4. 5 to the 1/2? Well, we can't take the square root of that, so let's just write that as the square root of 5. r to the fourth to the 1/2. There's two ways you can think about it. 4 times 1/2 is 2. So this is r squared. Or you could say the square root of r to the fourth is r squared. So this is r squared. Similarly, the square root of s to the fourth or s to the 1/2 is also s squared. And then this s to the 1/2, let's just write that as the square root of s. Just like that. Let's see what else we can do here. Let me write these other terms. We have an r to the 4/3 times s to the sixth times 2 times square root of 5 times r squared times s squared times the square root of s. Now, a couple of things we can do here. We could combine these s terms. Let's do that. Actually, just write the 2 out front first. So let's write the 2 out front first. So you have 2 times. Now let's look at these two s terms over here. We have s to the sixth times s squared. When someone says to simplify it, there's multiple interpretations for it. But we'll just say s to the sixth times s squared. That's s to the eighth. 6 plus 2. Times s to the eighth power. Times-- now this one's interesting and we might want to break it up depending on what we consider to be truly simplified. We have r to the 4/3 times r squared. r to the 4/3 is the same thing as r to the 1 and 1/3. That's what 4/3 is. So 1 and 1/3 plus 2 is 3 and 1/3. So we could write this times r to the 3 and 1/3. That's a little inconsistent. Over here I'm adding a fraction. Over here with the s I kind of left out the s to the 1/2 from the s's here. But we could play around with it and all of those would be valid expressions. So we've already dealt with the 2. We've already dealt with these two s's. We've already dealt with these r's. And then you have the square root of 5 times the square root of s. And we could merge them if we want, but I won't do it just yet. Times the square root of 5 times the square root of s. Now there's two ways we could do it. We might not like having a fractional exponent here. And then we could break it out. Or we might want to take this guy and merge it with the eighth power. Because you know that this is the same thing as s to the 1/2. So let's do it both ways. So if we wanted to merge all of the exponents, we could write this as 2 times s to the eighth times s to the 1/2. So s to the eighth and s to the 1/2. That would be 2 times s to the 8-- I can even write it as a decimal. 8.5. 8 plus-- you could imagine this is s to the 0.5 power. So that's 8.5 times r to the 3 and 1/3. I'm kind of mixing notations here. I have just a decimal notation, then I have a fraction notation, mixed number notation. Times the square root of 5. This is one simplification. I kind of have it in the fewest terms possible. The other simplification if you don't want to have these fractional exponents out here, you could write it as-- I'll do this in a different color. You could write this-- and these are all equivalent statements. So it's up to debate what simplified really means. So you could write this as 2 times s to the eighth. Instead of writing r to the 3 and 1/3, we could write r to the third times the cube root of r, which is the same thing as r to the 1/3. We could write r to the third times r to the 1/3. r to the 1/3 is the same thing as the cube root of r. And then you have the square root of these two guys. Both of these guys are being raised to the 1/2 power. So you could then say times the square root of 5s. I like this one a little bit more, the one on the left. To me this is really simplified. We've merged all of the bases. We have these two numbers, we've merged all the s terms, all the r terms. This is a little bit more complicated. You have a cube root. You haven't separated the s's and the r's. So I would go with this one if someone really wanted me-- said hey, Sal, simplify it how you like.