Suppose $a$, $b$, $n$ are positive. Prove that $a^n\mid b^n$ if and only if $a\mid b$.
I know that this can be proved through prime factorization, but I want to prove it using other methods.
I understand that the if and only if statement requires you to prove the two cases:
1. If $a^n\mid b^n$ then $a\mid b$.
2. If $a\mid b$ then $a^n\mid b^n$.
but I have no idea on how to approach it.
$a|b \ :\Leftrightarrow \exists k\in\mathbb{N} : k \cdot a = b$
1.) $$a^n|b^n \ \ \ \Rightarrow \ \ \ b^n = \hat{k} \cdot a^n $$
$$\sqrt[n]{\hat{k}} = \frac{b}{a}$$
Since $a,b$ are positive integers, $\frac{b}{a} \in \mathbb{Q}$. For $n>1$, the $n$-th root of a positive non-zero rational number is either an integer or irrational, but in this case it cannot be irrational because the RHS is rational. Therefore, $k' := \sqrt[n]{\hat{k}} \in \mathbb{N}$ and $b = k' \cdot a$, thus $a|b$.
2.) $$a|b \ \ \ \Rightarrow \ \ \ b = k \cdot a \ \ \ \Rightarrow \ \ \ b^n = k^n \cdot a^n \ \ \ \Rightarrow a^n | b^n$$ because $k^n \in \mathbb{N}$.
P.S.: The approach for 1.) is beating around the bush, you should really use the prime factorization as the main argument.