Is $\frac{\sqrt7}{\sqrt[3]15}$ rational or irrational? Prove it.
I am having a hard time with this question. So far what I did was say, assume it's rational, then $$\frac{\sqrt7}{\sqrt[3]15}=\frac{x}{y} \Rightarrow \sqrt{7}y=x\sqrt[3]15$$
I then showed the product of a rational number and an irrational number is irrational so the expression above is irrational on the left and right side. I can't get to a contradiction to prove it's irrational so I'm currently thinking it is rational but I don't know how to go about proving it.
Answer
Factoring $\,7^{\,\large \color{}{3}} y^{\large 6}\!=15^{\large 2}x^{\large\color{}{6}}$ into primes, $\,7\,$ has odd power $\,3\!+\!6j\,$ on LHS, but even $\,6k\,$ on RHS
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