How To Add Fractions With Square Roots In The Denominator

we're asked todivide and simplify. and we have oneradical expression over another radical expression. the key to simplifythis is to realize if i have the principal root ofx over the principal root of y,

How To Add Fractions With Square Roots In The Denominator, this is the same thing as theprincipal root of x over y. and it really just comes outof the exponent properties. if i have two things thati take to some power-- and taking the principal rootis the same thing as taking it

to the 1/2 power-- ifi'm raising each of them to some power andthen dividing, that's the same thing asdividing first and then raising them to that power. so let's apply that over here. this expressionover here is going to be the same thingas the principal root-- it's hard to writea radical sign that big-- the principal rootof 60x squared y over 48x.

and then we can first lookat the coefficients of each of these expressions andtry to simplify that. both the numerator and thedenominator is divisible by 12. 60 divided by 12 is 5. 48 divided by 12 is 4. both the numerator and thedenominator are divisible by x. x squared dividedby x is just x. x divided by x is 1. anything we dividethe numerator by,

we have to dividethe denominator by. and that's all we have left. so if we wantedto simplify this, this is equal to the--make a radical sign-- and then we have 5/4. and actually, we can write itin a slightly different way, but i'll write itthis way-- 5/4. and we have nothing left in thedenominator other than that 4. and in the numerator, wehave an x and we have a y.

and now we could leaveit just like that, but we might want to take morethings out of the radical sign. and so one possibilitythat you can do is you could say that this isreally the same thing as-- this is equal to 1/4 times 5xy, allof that under the radical sign. and this is the samething as the square root of or the principalroot of 1/4 times the principal root of 5xy. and the square root of1/4, if you think about it,

that's just 1/2 times 1/2. or another way youcould think about it is that this right hereis 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 realizeit's 1/2, you say, hey, this is the same thingas the square root of 1 over the square root of 4,and the square root of 1 is 1 and the principal root of 4 is2, so you get 1/2 once again.

and so if you simplifythis right here to 1/2, then the whole thing cansimplify to 1/2 times the principal root--i'll just write it all in orange-- times theprincipal root of 5xy. and there's nothingelse that you can really take out of theradical sign here. nothing else hereis a perfect square.