If 1953125 is a square number, then [tex]1953125=n^2,~n\in\mathbb{N}[/tex]
By what's given, [tex]1953125=5^9\Rightarrow n^2=5^9\Rightarrow n=5^{4.5}=625\sqrt{5}[/tex]
But [tex]n=625\sqrt{5} \notin\mathbb{N}[/tex], therefore, our assumption that 1953125 is a square number is wrong.
