$k$ be a field , then is it true that $k[x]/(x^m) \otimes _k k[x]/(x^n) $ is isomorphic to $k[x,y]/(x^m , y^n)$ as $k$-algebra s?

Question

Let $k$ be a field , then is it true that $k[x]/(x^m) \otimes _k k[x]/(x^n) $ is isomorphic to $k[x,y]/(x^m , y^n)$ as $k$-algebra s ?

Unfortunately I am not able to apply any of the properties of tensor products here . Please help . Thanks in advance