(1+sixthroot2 - twelfthroot2) to the -1 power

Answer 1

(1 + 2^1/6 - 2^1/12)^-1
1/(1 + 2^1/6 - 2^1/12)
1/[(2^11/12 + 2^1/12 - 1)(2^1/12)]

That can be factored somewhat, but it won't get any simpler than what you started with.