linear independent over $\mathbb{Q}$

We may without loss of generality assume that

$i < j \Longrightarrow r_i < r_j; \tag 0$

since $0 < r_i \in \Bbb Q$, $1 \le i \le n$, we may write

$r_i = \dfrac{p_i}{q_i}, \; p_i, q_i \in \Bbb Z, 0 < p_i < q_i; \tag 1$

we choose $m = \text{lcm} \{q_i \mid 1 \le i \le n \}, \tag 2$

that is, $m$ is the least common multiple of the denominators $q_i$; as such, it is the least common denominator of the $r_i = p_i / q_i$, and so each $r_i$ may be written

$r_i = \dfrac{k_i}{m}, \; 0 < k_i \in \Bbb Z; \tag 3$

we note that (0) implies $k_i < k_j$ for $i < j$; if the numbers $2^{r_i}$ are linearly dependent over $\Bbb Q$, we have $\alpha_i \in \Bbb Q$, not all $0$, with

$\displaystyle \sum_1^n \alpha_i 2^{r_i} = 0; \tag 4$

we observe that at least two of the $\alpha_i \ne 0$, lest $2^{r_i} = 0$ for some $i$, impossible; using (3), (4) may be written

$\displaystyle \sum_1^n \alpha_i 2^{k_i / m} = 0; \tag 5$

since $r_i \in (0, 1)$ for all $i$, we have $k_1 > 0$ and $k_n < m$; thus we may factor $2^{k_1/m}$ out of (5) and, setting $t_i = k_i - k_1$, so that $t_1 = 0$ and $t_n < m$, write

$\displaystyle \sum_1^n \alpha_i 2^{t_i / m} = 0; \tag 6$

let

$f(x) = \displaystyle \sum_1^n \alpha_i x^{t_i} \in \Bbb Q[x]; \tag 7$

then (6) affirms that

$f(2^{1/m}) = 0; \tag 8$

we also have

$\deg f(x) \le t_n < m; \tag 9$

it is evident that $r=2^{1/m}$ satisfies

$g(x) = x^m - 2 \in \Bbb Q[x]; \tag{10}$

Eisenstein with $p = $ shows that $g(x)$ is irreducible over $\Bbb Q$; hence $2^{1/m}$ can satisfy no polynomial of degree less than $\deg g(x) = m$, but this contradicts (8), whence there can be no linear dependence such as (4). And we are done.

Note: in retrospect, having written up this proof, the division through by $2^{k_1/m}$ and the introduction of the $t_i$ don't really appear necessary; I must have been thinking along a different track when I introduced them; nevertheless, this doesn't affect the validity of the result, and it's less work to leave those steps in place than edit then out. End of Note.