$K$ field of characteristic $p$, where $p$ is prime Is there a way I can prove $(a+b)^{p^n} = a^{p^n} + b^{p^n}$, where int $n \ge 0$, $a,b\in K$ and $p$ prime the characteristic of $K$?
Can I use the binomial theorem and proof by induction? Both or just one?

You will want both, the binomial theorem says $p\big| {p\choose k}$ when $1\le k\le p-1$ so that
$$(a+b)^p\equiv a^p+b^p\mod p$$
Then using induction you see that
$$(a+b)^{p^n} = \big((a+b)^{p^{n-1}}\big)^p = (a^{p^{n-1}}+b^{p^{n-1}})^p = a^{p^n}+b^{p^n}.$$
