How do you simplify sqrt(x^3y^3 )/sqrt(xy)x3y3xy?

1 Answer
Oct 1, 2015

sqrt(x^3*y^3)/sqrt(x*y) = x*yx3y3xy=xy
for all x != 0x0 and y != 0y0
and it's undefined if either x=0x=0 or y=0y=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=0x=0 and y=0y=0 as those, when the expression is undefined since its denominator would by 00.

For all other cases, when x != 0x0 and y != 0y0 we transform the expression as follows:

sqrt(x^3*y^3)/sqrt(x*y) = sqrt(x^2*y^2*x*y)/sqrt(x*y) = x3y3xy=x2y2xyxy=
= sqrt(x^2*y^2)*sqrt(x*y)/sqrt(x*y) = =x2y2xyxy=
= sqrt((x*y)^2)*sqrt(x*y)/sqrt(x*y) = =(xy)2xyxy=
= x*y*sqrt(x*y)/sqrt(x*y) = x*y*1 = x*y=xyxyxy=xy1=xy