Simplifying radical expressions: three variables

We're asked to simplify the cube root of 27a squared times b to the fifth times c to the third power. And the goal, whenever you try to just simplify a cube root like this, is we want to look at the parts of this expression over here that are perfect cubes, that are something raised to the third power. Then we can take just the cube root of those, essentially taking them out of the radical sign, and then leaving everything else that is not a perfect cube inside of it. So let's see what we can do. So first of all, 27-- you may or may not already recognize this as a perfect cube. If you don't already recognize it, you can actually do a prime factorization and see it's a perfect cube. 27 is 3 times 9, and 9 is 3 times 3. So 27-- its prime factorization is 3 times 3 times 3. So it's the exact same thing as 3 to the third power. So let's rewrite this whole expression down here. But let's write it in terms of things that are perfect cubes and things that aren't. So 27 can be just rewritten as 3 to the third power. Then you have a squared-- clearly not a perfect cube. a to the third would have been. So we're just going to write this-- let me write it over here. We can switch the order here because we just have a bunch of things being multiplied by each other. So I'll write the a squared over here. b to the fifth is not a perfect cube by itself, but it can be expressed as the product of a perfect cube and another thing. b to the fifth is the exact same thing as b to the third power times b to the second power. If you want to see that explicitly, b to the fifth is b times b times b times b times b. So the first three are clearly b to the third power. And then you have b to the second power after it. So we can rewrite b to the fifth as the product of a perfect cube. So I'll write b to the third-- let me do that in that same purple color. So we have b to the third power over here. And then it's b to the third times b squared. So I'll write the b squared over here. And we're assuming we're going to multiply all of this stuff. And then finally, we have-- I'll do in blue-- c to the third power. Clearly, this is a perfect cube. It is c cubed. It is c to the third power. So I'll put it over here. So this is c to the third power. And of course, we still have that overarching radical sign. So we're still trying to take the cube root of all of this. And we know from our exponent properties, or we could say from our radical properties, that this is the exact same thing. That taking the cube root of all of these things is the same as taking the cube root of these individual factors and then multiplying them. So this is the same thing as the cube root-- and I could separate them out individually. Or I could say the cube root of 3 to the third b to the third c to the third. Actually, let's do it both ways. So I'll take them out separately. So this is the same thing as the cube root of 3 to the third times the cube root-- I'll write them all in. Let me color-code it so we don't get confused-- times the cube root of b to the third times the cube root of c to the third times the cube root-- and I'll just group these two guys together just because we're not going to be able to simplify it any more-- times the cube root of a squared b squared. I'll keep the colors consistent while we're trying to figure out what's what. And I could have said that this is times the cube root of a squared times the cube root of b squared, but that won't simplify anything, so I'll just leave these like this. And so we can look at these individually. The cube root of 3 to the third, or the cube root of 27-- well, that's clearly just going to be-- I want to do that in that yellow color-- this is clearly just going to be 3. 3 to the third power is 3 to the third power, or it's equal to 27. This term right over here, the cube root of b to the third-- well, that's just b. And the cube root of c to the third, well, that is clearly-- I want to do that in that-- that is clearly just c. So our whole expression has simplified to 3 times b times c times the cube root of a squared b squared. And we're done. And I just want to do one other thing, just because I did mention that I would do it. We could simplify it this way. Or we could recognize that this expression right over here can be written as 3bc to the third power. And if I take three things to the third power, and I'm multiplying it, that's the same thing as multiplying them first and then raising to the third power. It comes straight out of our exponent properties. And so we can rewrite this as the cube root of all of this times the cube root of a squared b squared. And so the cube root of all of this, of 3bc to the third power, well, that's just going to be 3bc, and then multiplied by the cube root of a squared b squared. I didn't take the trouble to color-code it this time, because we already figured out one way to solve it. But hopefully, that also makes sense. We could have done this either way. But the important thing is that we get that same answer.