Power Divides Power

Since $a^{n} \mid b^{n}$ , $\frac{b^{n}}{a^{n}}=(\frac{b}{a})^{n}$ is an integer. Then $\frac{b}{a}$ is also an integer because integral root of integer is either an irrational number or an integer, which means that $a \mid b$ .

it is true that 2^2 divides 4^2 then let it be true for a^n divides b^n then a^(n+1)/b^(n+1) =a^n a/b^n b =a a a a a... a/b b b b b... b is divisible. Then by mathematical induction it is true for all values of a,b