Our first step in simplifying this radical expression is to utilize the fact that, for any two numbers #a# and #b#:
#sqrt(ab)=sqrtasqrtb#
With this in mind, we can rewrite the original expression as
#sqrt(18xy^2)/sqrt(49x)=(sqrt18sqrtxsqrt(y^2))/(sqrt49sqrtx)#
We can then cancel the #sqrtx# in the numerator and the denominator to obtain
#(sqrt18cancel(sqrtx)sqrt(y^2))/(sqrt49cancel(sqrtx))=(sqrt18sqrt(y^2))/sqrt49#
The square root of any number squared is simply that number, so we can simplify #sqrt(y^2)# to #y#, and since #49# is just #7^2#, #sqrt49# must be #7#. We now have:
#(sqrt18y)/7#
If we want to simplify #sqrt18#, we'll need to factor a perfect square out of the #18#, and then again take advantage of the property of radicals stated at the beginning of the explanation.
Since #18# is the product of the perfect square #9# and #2#, we can rewrite the radical as #sqrt(9*2)#, which is equivalent to #sqrt9sqrt2#. Since #9# is #3^2#, we can simplify #sqrt9# to #3#; we can't simplify #sqrt2# any more, so our simplified radical for #sqrt18# would be #3sqrt2#.
Putting that back into our expression, we finally get #(3sqrt2y)/7#, which, depending on your aesthetic taste, you can either leave as-is or rewrite as #(3ysqrt2)/7#.
