Equivalence relation $x^m=y^n$

Question

Show that $R=\{(x,y) \in\mathbb{N}^2:\exists m,n \in \mathbb{N} \text{ s.t. } x^m=y^n\}$ is an equivalence relation or disprove otherwise

Reflexivity and symmetricity were really easy to show but how do I show that it's transitive?

Let $(a,b),(b,c)\in R$, so $a^m=b^n$ and $b^{m'}=c^{n'}$, how can I show that $a^{m''}=c^{n''}$?

Answer 1

Let $a^m = b^n$ and $b^x = c^y$.
Then $(a^m)^x = (b^n)^x$ and $(b^x)^n = (c^y)^n$.
We have $a^{mx} = b^{nx} = c^{yn} \implies a^{mx} = c^{yn}$

Answer 2

In your notation, try $m'' = mm'$ and $n'' = nn'$.