Is it possible that some combination of partial sums of $\sum \sqrt{k}$ be a rational number? $\text{where } k \text { not a perfect square}$

Is it possible that some combination of partial sums of $\sum \sqrt{k}$ be a rational number? $k=2,3,5,\cdots \text{where } k \text { not a perfect square}$

More ever : can a linear combination of $\sqrt[m_1]{k_1}$ , $\sqrt[m_2]{k_2}, \cdots$ be rational , where $k_i$ integer, is not perfect power of $m_i$? I.e. $k_i \neq a^{m_i}$ from some integer $a_i$?