Simplifying radical expressions: three variables

we're asked to simplify the cube root of 27 a squared times B to the fifth times C to the third power and the goal whenever you try to just simplify a cube 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 3rd power that 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 try 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 B to the fifth is not a perfect cube by itself but kit can be expressed as a 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 this 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 a product of a perfect cube so I'll write B to the third I'm doing 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 we're assuming we're going to multiply all of this stuff and then finally finally we have hundreds in blue C to the third power clearly this is a perfect cube it is C cubed it is see 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 three to the third B to the third C to the third actually let's do it both ways so I'll set take them out separately so this is the same thing as the cube root of three to the third times the cube root I'll write them all in well right dude let me color code it so we don't get confused times the cube root of B to the third times the cube root times the cube root of C to the third 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 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 a squared B squared and I could have I could have said that this is times the cube root of a squared times 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 right 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 that's just B and the cube root of C to the third 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 three times B times C times C times the cube root times the cube root of a squared B squared times the cube root of a squared a squared the squared and we're done I just want to do one other thing just because I did mention that I would do it we could simplify this way or we could recognize we could recognize that this expression right over here can be written as 3 B C 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 come straight out of our exponent properties and so we can rewrite this as the cube root the cube root of all of this times the cube root times the cube root of a-squared b-squared and so the cube root of all of this of 3b C to the third power well that's just going to be three BC and then multiplied by the cube root of a-squared b-squared I didn't take the trouble to color code at this time because we already figured out one way to solve it but hopefully that also makes sense we could have gone we could have done this either way but the important thing is is that we get that same answer