What can be said about two rational but transcendental numbers

Let $a,b$ be distinct rational numbers such that $a^n=b^n$ for some $n$
(note that this does not necessarily mean that $a=b$).
My question is, what can be said about the set $X=\{a,b\}$?

Question. Is $X$ dense in $\mathbb Q$? What is the minimal $n$ such that $a^n=b^n$?

I am interested in the general case (i.e., there may not be a solution) and whether or not this has an impact on the answer.

A:

Hint: $a$ and $b$ are linearly independent over $\mathbb{Q}$, so pick some pair of integers $m$ and $n$ such that $ma+nb=0$. Can you finish?

A:

If $a e b$, then there exists $\frac{p_1}{q_1}\in\mathbb{Q}$ such that $a-b=\frac{p_1}{q_1}$.
Now $a^n-b^n=(a-b)\cdot\left(a^{n-1}+a^{n-2}b+\ldots+ab^{n-2}+b^{n-1}\right)$
So we can find rational $p_2,q_2$ such that
$a^n-b^n=(p_1p_2)$
$(a-b)a^{n-1}=-p_1p_2+q_1q_2$
\$(a-
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