Is $12^{1/3}$ irrational?

Question

Is $12^{1/3}$ irrational? Give a proof that justifies your answer

So far I have: Suppose $12^{1/3}$ is rational.This means there exists integers a and b such that $12^{1/3} = \frac{a}{b}$ where $\gcd(a,b) = 1$.

Then, $12 = \frac{a^3}{b^3}$ so $a^3 = 12b^3$. This means $12|a^3$. However since $12$ is not prime, you can't say $12|a$.

Please help!

Answer 1

Note that $3|a^3$, so $3|a$. So then $3|b^3$ and $3|b$. Can you take it from here?