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At my latest exam was the following problem:

Prove that $\sqrt{2}+\sqrt[3]{2}$ is irrational.

The solution is to find a polynomial with integer coefficients, a root of which is $\sqrt{2}+\sqrt[3]{2}$ and to prove it is not a rational root of it. How do you find polynomials for similar expressions (like the general case of $\sqrt[k]{m}+\sqrt[l]{n}$, given $\sqrt[k]m, \sqrt[l]n \not\in \mathbb{Q}$)?

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  • The title question with concrete numbers has been asked and answered before. There are tips for the more general version as well. Once you have a candidate minimal polynomial you can verify that it has no rational roots using the rational root test. Mind you, there are other methods for getting irrationality of numbers like this. – Jyrki Lahtonen May 18 '19 at 11:06

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