Show that $\forall n\in \mathbb{N}$, the function $e^{-x^n}$ is of exponential order and its Laplace transform exists on $[0,\infty)$

Question

Show that $\forall n\in \mathbb{N}$, the function $e^{-x^n}$ is of exponential order and its Laplace transform exists on $[0,\infty)$

So we need to show that $e^{-sx} |f(x)|$ converges to show that it is of exponential order.

$e^{-sx}*e^{-x^n}$ converges to 0 as x approaches infinity so it is of exponential order.

How do I go about saying that its Laplace transformable though?