Let $\ast$ denote the "geometric mean" operation: $a\ast b=\sqrt{ab}$ for positive numbers $a,b$. Which of the following distributive laws hold for all positive numbers $x$, $y$, and $z$?

I. $x\ast (y+z)=(x\ast y)+(x\ast z)$
II. $x+(y\ast z)=(x+y)\ast (x+z)$
III. $x\ast (y\ast z)=(x\ast y)\ast (x\ast z)$