Proof that $(ab+cd)^{\frac{1}{n}}$ is irrational?

Question

Let $a,b,c,d,n >2, \gcd(a,b,c,d)=1$, how can I prove that $\sqrt[n]{ab+cd}$ is irrational if $\sqrt[n]{a},\sqrt[n]{b},\sqrt[n]{c},\sqrt[n]{d}$ are irrational? Any hint?

Answer 1

It may not be true.

Consider for example $a=3, b=5, c=53, d=77$. Then $$\sqrt[3]{3\times5+53\times77} =\sqrt[3]{4096}=16.$$

There are probably simpler examples.