Why doesn't $\sqrt {x^3} = x \sqrt {x}$

Question

According to the law of radicals:

$$ \sqrt {an} = \sqrt a \cdot \sqrt n $$

Wouldn't it make sense that: $\sqrt {x^3}= \sqrt {x^2} \cdot \sqrt x = x \sqrt x$

Obviously this doesn't work if you plug in values and compare, but this makes logical sense?

Answer 1

Your formula is mostly correct, except that it should read $$\sqrt {x^3}= \sqrt {x^2} \cdot \sqrt x = \color{red}{|x| \sqrt x}$$