Does $a^n \mid b^n$ imply $a\mid b$? Does $a^n \mid b^n$ imply $a\mid b$? I think it does but haven't been able to prove it. 
I don't know much number theory so an elementary answer would be great.

Hint $\ $ Either examine exponents in unique prime factorizations, or, by the Rational Root Test, the reduced rational root $\rm\:x = b/a\:$ of $\rm\:x^n = c\in\Bbb Z\:$ must be integral, so $\rm\:b/a\in\Bbb Z\:\Rightarrow\:a\mid b.$
