I substituted $n=\frac 1{x^2}$, which also means that $x^2=\frac 1 n$.
So we get $\Big(1 + \frac 32 x^2\Big)^{1/x^2} = \Big(1 + \frac 32 \cdot \frac...
I have obtained access to the full solutions manual, after contacting Wiley about it. I will not type up the solution in full, but simply note a few...