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.