Simplifying $\sqrt{xy\mathstrut}\sqrt{x^3y}$ - two different paths, different results I'm trying to simplify $\sqrt{xy\mathstrut}\sqrt{x^3y}$, for which the book has the solution below:
$$\sqrt{xy\phantom{\big|}}\sqrt{x^3y} = \sqrt{(xy)(x^3y)} = \sqrt{x^4y^2} = x^2|y|$$
I understand and agree with the above solution.  That said, prior to looking at the solution, I did the following, based on $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$:
$$\sqrt{xy\phantom{\big|}}\sqrt{x^3y}$$
$$\sqrt{xy\phantom{\big|}}\sqrt{x\cdot y\cdot x^2}$$
$$\sqrt{xy\phantom{\big|}}\sqrt{xy\phantom{\big|}}\sqrt{x^2}$$
$$\sqrt{(xy)^2}\sqrt{x^2}$$
$$xy\sqrt{x^2}$$
$$xy|x|$$
Clearly, I'm getting a different/wrong answer.  I think I'm following the rules of radicals but am not getting the same result.  

What am I doing wrong?  


$$\sqrt{(xy)^2}\sqrt{(x^2)} = |xy|\sqrt{(x^2)} = |xy|\cdot |x| = |x|\cdot|y|\cdot |x| =  |x|^2\cdot|y| = x^2|y|$$
