Which irrationals become rational for some positive integer power? Related to Irrationals becoming rationals after being raised to some power. Let $r \in \mathbb{R} \setminus \mathbb{Q}$. True or false: there exists an $n \in \mathbb{N}$ (positive integers) such that $r^n = r \cdot \dots \cdot r \in \mathbb{Q}$. This is clearly true for some irrationals like $\sqrt{2}$ or $a^{1/n}$ (positive integer $a$ such that $a^{1/n} \notin \mathbb{Z}$; see How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?). But is it true for ALL irrationals? If not, can we classify all the irrationals for which the statement is true?

You have to find the transcendental numbers like exponential number, π, sin(a) , sin h(a) etc. which make your statement wrong! One thing that you have to remind, every transcendental numbers are irrational numbers, but converse is not always true i.e, there can be irrationals which are not transcendental numbers, like the numbers what you have given and these are part of algebraic numbers, which are suitably described in 1st answer!
So, for your question, only algebraic irrational numbers can be transformed into rational numbers by repeated multiplication as many times you need. But transcendental irrational numbers can't be!
Also cardinality of the set of these transcendental irrational numbers is uncountable!
But on the other hand, cardinality of the set of these algebraic rational numbers is countable!
