Question: Assume $x\gt 0$, how does one simplify $$e^{(-x^2t)/i}\ ?$$

I don't understand how we could change the i under to the top so I could use Euler's formula

Answer: 
Remember that $-i^2 = 1$ implies that $-i = \frac{1}{i}.$ Then apply Euler's formula as you mention.
$\frac{1}{i}=-i$. In general, $\frac{1}{a+bi}=\frac{a-bi}{a^2+b^2}$. You can probably figure out the rest.