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we're asked to divide and simplify and we have one radical expression over another radical expression the key to simplify this is to realize if I have the square the principal root of x over the principal root of y this is the same thing as the principal root of x over Y and it really just comes out of the exponent properties if I have two things but I take to some power and picking the principal root is the same thing as taking it to the one-half power if I'm raising each of them to some power and then dividing that's the same thing as dividing first and then raising them to that power so let's apply that over here this expression over here is going to be the same thing as the principal root the principal it's hard to write a radical sign that big the principal root of 60 x squared Y over 48 X and then we can first look at the coefficient terms or the coefficients of each of these expressions and try to simplify that both the numerator and the denominator is divisible by 12 60 divided by 12 is 5 48 divided by 12 is 4 both the numerator and the denominator is divisible by X x squared divided by X is just X X divided by X is 1 anything we divide the numerator by we have to divide the denominator by and that's all we have left so if we wanted to simplify this this becomes this is equal to the the make a radical sign and then we have 5/4 5/4 5/4 and actually we can write it a little slightly different way but I'll write it this way 5/4 and we have nothing left in the denominator other than that 4 and in the numerator we have an X and we have a Y we have an X and we have a Y and now we could leave it just like that but we might want to take more things out of the radical sign and so one possibility that you can do is you could say that this is really the same thing as this is equal to 1/4 times 5 5 X Y all of that under the radical sign and this is the same thing as the square root of or the principal root of 1/4 times the principal root of 5x y and the square root of 1/4 if you think about it that's just 1/2 times 1/2 or another way you could think about it is this is the same thing this right here is the same thing as so you could just say hey this is 1/2 1/2 times 1/2 is 1/4 or if you don't realize it's 1/2 you say hey this is the same thing as the square root of 1 over the square root of 4 and the square root of 1 is 1 and square root the principal root of 4 is 2 so you get 1/2 once again and so if you simplify this right here to 1/2 then the whole thing can simplify to 1/2 times the principal root I'll just write it all in orange times the principal root of 5 X Y and there's nothing else that you can really take out of the radical sign here nothing else here is a perfect square