Question

The number 1953125 is equal to 5^9. explain how the information above informs you that 1953125 is not a square number

Answer 1

If 1953125 is a square number, then $1953125=n^2,~n\in\mathbb{N}$

By what's given, $1953125=5^9\Rightarrow n^2=5^9\Rightarrow n=5^{4.5}=625\sqrt{5}$

But $n=625\sqrt{5} \notin\mathbb{N}$, therefore, our assumption that 1953125 is a square number is wrong.