Let
Prove that
For example
show that (b1 sqrt(a1) + b2(sqrt(a2)) ( b1 sqrt(a1) - b2 sqrt(a2)) (-b1 sqrt(a1) + b2 sqrt(a2)) (-b1 sqrt(a1) - b2 sqrt(a2)) is rational.
A hand waving argument is as follows.
For each b value, b1, b2, b3,....bk,
then the b is multiplied by a different b,
for example b1 * b2, it is , in another term,
b1 is multiplied by another -b2.
The basic idea is that all possible signs are assigned to the b values.
so all the terms of b1 sqrt(a1) with a different b+j sqrt(a_j) will add to zero.
this leaves the product to be
b1^(2^k) a1^(2^(k-1)) b2^(2^k) a2^(2^(k-1)) ....
which is rational.