Prove expressions rational

[elim]

Let \(B_n = \{-1,1\}^n, \; a_k \in \mathbb{Q}, \; k=1,\cdots,n\)
Prove that
(1) \(\sum_{(b_1,\cdots,b_n)\in B_n} (b_1 \sqrt{a_1}+ \cdots +b_n \sqrt{a_n})^{2m} \in \mathbb{Q}, \; m \in \mathbf{N}\)
(2) \(\prod_{(b_1,\cdots,b_n)\in B_n} (b_1 \sqrt{a_1}+ \cdots +b_n \sqrt{a_n}) \in \mathbb{Q}\)