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 $5$.