Simplifying $\sqrt{xy\mathstrut}\sqrt{x^3y}$ - two different paths, different results

Question

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?

Answer 1

At the stage where you simplify $\sqrt{(xy)^2}$ you should get $|xy|$, not $xy$. Therefore your final line should be $|xy||x|$, which is equal to $|x|^2 |y| = x^2 |y|$

Answer 2

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

Answer 3

The previous answers are right, but it may be of interest that your answer, though incorrectly arrived at, is in fact correct. The expression makes no sense unless $x$ and $y$ have the same sign, and then your answer is equivalent to the one in the book.