Question

How do you simplify `sqrt(x^3y^3 )/sqrt(xy)`?

Answer 1

`sqrt(x^3*y^3)/sqrt(x*y) = x*y`
for all `x != 0` and `y != 0`
and it's undefined if either `x=0` or `y=0`

Explanation:

It's essential, before transforming any algebraic expression, to determine its domain, because during transformations we might derive with seemingly equivalent expression that has a different domain, and we will not have the right to say that original and final expressions are equivalent.

In this case we should exclude values `x=0` and `y=0` as those, when the expression is undefined since its denominator would by `0`.

For all other cases, when `x != 0` and `y != 0` we transform the expression as follows:

`sqrt(x^3*y^3)/sqrt(x*y) = sqrt(x^2*y^2*x*y)/sqrt(x*y) =`
`= sqrt(x^2*y^2)*sqrt(x*y)/sqrt(x*y) =`
`= sqrt((x*y)^2)*sqrt(x*y)/sqrt(x*y) =`
`= x*y*sqrt(x*y)/sqrt(x*y) = x*y*1 = x*y`