What remarks about the tails can we infer from the density function of $X^n$ if $X$ is standard normally distributed

Question

1. Find the pdf of $X^n$ if $X\sim N(0,1)$

Is there any general formula for this or we have to divide into 2 cases: $n$ is an odd number; and $n$ is an even number?

2. What can we learn about the tails of the densities when $n$ receives different values?

I hope to find some help. Thank you.