Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 2 Exercises 2.1 & 2.2

Our main tool is Corollary 2.3.

Solution: We only show that $\sqrt[5]{7}$ is irrational.

Clearly, $\sqrt[5]{7}$ is a solution of $x^5-7=0$. Hence by Corollary 2.3, the rational solution to this equation can only be $\pm 1$ and $\pm 7$. It is straightforward to check that none of them are solutions of $x^5-7=0$. Hence $x^5-7=0$ does not have rational solutions and $\sqrt[5]{7}$ is irrational.

The rests are similar.