For some reason I cannot figure out how the book is finding the solution to this problem.

find the inverse function of $f(x)=3x\sqrt{x}$.

My steps seem to lead to a dead end:

step 1. switch $f(x)$ with $y$: $y = 3x\sqrt{x}$

step 2. swap $x$ and $y$: $x = 3y\sqrt{y}$

step 3. solve for $y$: 

step 3.1: $\displaystyle \frac{x}{3y} = \sqrt{y}$.

step 3.2: $\displaystyle \left(\frac{x}{3y}\right)^2 = \left(\sqrt{y}\right)^2$

step 3.3: $\displaystyle \frac{x^2}{9y^2} = y$ ... uhhh?

The book answer is $\displaystyle y = \left(\frac{x}{3}\right)^{2/3}$.