Let $\displaystyle B_n = \{-1,1\}^n = \{(b_1,\cdots,b_n) \mid b_k \in \{-1,1\}, k=1,\cdots,n\}, \; a_k \in \mathbf{Q},\; a_k > 0, \; \forall k$

Prove that $\displaystyle \prod_{(b_1,\cdots,b_n)\in B_n}(b_1 \sqrt{a_1}+\cdots + b_n \sqrt{a_n}) \in \mathbf{Q}$