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# Simplifying radical expressions: two variables

A worked example of simplifying elaborate expressions that contain radicals with two variables. In this example, we simplify √(60x²y)/√(48x). Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

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 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 that
I take to some power-- and taking the principal root
is the same thing as taking it to the 1/2 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-- it's hard to write
a radical sign that big-- the principal root
of 60x squared y over 48x. And then we can first look
at 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 are 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 is equal to the--
make a radical sign-- and then we have 5/4. And actually, we can write it
in a 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. 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 5xy, 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 5xy. 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 that 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 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 5xy. And there's nothing
else that you can really take out of the
radical sign here. Nothing else here
is a perfect square.