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

Oct 1, 2015

$\frac{\sqrt{{x}^{3} \cdot {y}^{3}}}{\sqrt{x \cdot y}} = x \cdot y$
for all $x \ne 0$ and $y \ne 0$
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 $x \ne 0$ and $y \ne 0$ we transform the expression as follows:

$\frac{\sqrt{{x}^{3} \cdot {y}^{3}}}{\sqrt{x \cdot y}} = \frac{\sqrt{{x}^{2} \cdot {y}^{2} \cdot x \cdot y}}{\sqrt{x \cdot y}} =$
$= \sqrt{{x}^{2} \cdot {y}^{2}} \cdot \frac{\sqrt{x \cdot y}}{\sqrt{x \cdot y}} =$
$= \sqrt{{\left(x \cdot y\right)}^{2}} \cdot \frac{\sqrt{x \cdot y}}{\sqrt{x \cdot y}} =$
$= x \cdot y \cdot \frac{\sqrt{x \cdot y}}{\sqrt{x \cdot y}} = x \cdot y \cdot 1 = x \cdot y$