# Find natural number $0 < n < 30,000$ such that $\sqrt[3]{5n}+\sqrt{10n}$ is rational

I was thinking that I could try to make some sort of substitution to convert $$\sqrt[3]{5n}+\sqrt{10n}$$ into a polynomial with integer coefficients then use the Rational Roots Theorem to find a rational root. I don't really know if that's going to get me anywhere other than $$n=0$$.

I would really appreciate some help, or some hints. I don't necessarily want a full solution, but a nudge in the right direction.

You should focus on the prime factorization of $$n$$. To have a number be a cube, all the primes in its factorization need to come with a power that is a multiple of $$3$$. Similarly, to be a square the primes need to have an even power.
In your example, all primes except $$2$$ and $$5$$ must be sixth powers. Think about what the leading coefficients do for $$2$$ and $$5$$.
• By leading coefficients I mean the $5$ and $10$ under the radical signs. They mean you don't want the powers of $2$ and $5$ in $n$ to be multiples of $6$. – Ross Millikan Dec 9 '18 at 15:38
• I think the limit of $30,000$ is unfortunately low. I think it should have been chosen large enough for there to be half a dozen solutions or so. – Ross Millikan Dec 9 '18 at 15:56
• I just would like to have people recognize that once they find the smallest solution, which is $2^35^5=25,000$ they can multiply it by the sixth power of any number, including $2$ and $5$ and get another solution. – Ross Millikan Dec 9 '18 at 16:16